I am building a subreddit community and looking at similar subreddits for ideas on how to improve my user engagement. One thing that I have been noticing is that the smaller communities often have a rule that downvotes shouldn't be utilized unless the post is spam. One even had the downvote button disabled. Jumping over to Facebook, I realized they have a similar system - you can "Like" a post, but you can't "Dislike" it. Do negative actions against peer posts affect the health of a young (and small) community?
Downvotes control people. Beyond that, it's really your decision whether "control" is a good or bad thing.
Many decisions in the architecture and UX design of any service come in the distinction of whether it's trying to let people complete a specific task, and I think that same principle can be applied here. I'm basing that somewhat off of goal-oriented web design, but I'd rather not tie myself to that definition.
An example of such a task might be getting an answer to a question here on Community Building Stack Exchange. You come here with a question, and I come here with an answer. Each of us has the goal of sharing that information with the other. Yeah, I happen to be bored, dodging some work I have to do, so I'm poking around Stack Exchange as a whole, but fundamentally that's not what it's built for.
A counter-example might be a Facebook timeline. People go on Facebook with the fairly broad goal of "seeing what their friends are up to." Sure, it's a goal. But it's pretty subjective--you probably won't "finish" seeing what your friends are up to, so much as just decide you've seen enough.
It's important to control people and content when people have a tangible goal in your community, and downvotes are an effective way of doing that. Here on Stack Exchange, we leverage downvotes to help bring the best content to the top, and to help deter bad content from being posted. We put quality over quantity, in other words.
If I posted an answer here containing some nonsense (I hope what I'm actually posting isn't nonsense...) about how downvotes are implemented here on Stack Exchange to censor minorities, that wouldn't be beneficial to the community, nor you as an asker, so it would be good for that to be controlled.
For a community like Facebook, it's important to let people express themselves as freely as possible, within the ToS and local laws. Downvotes don't help with that. They would make people feel unsafe to share, which would hurt Facebook both as a business and as a community.
If I posted on Facebook about a new haircut and immediately received many downvotes, that probably wouldn't make me feel all that great about myself, my friends, or Facebook as a whole.
I don't think it has a whole lot to do with the size of the community. Community Building is still fairly small, but downvotes still play a pretty important aspect in the whole system of it. It's just a matter of what people want to get out of it, and what balance you want to strike between quality and expressiveness.
One thing to watch out for, is too much control. If you support downvotes, you need to make sure the scope of your community is well-defined and solid. Otherwise, you risk a small clique of users targeting certain borderline posts into oblivion, which would counter the growth of the community.
Particularly in that light, I can understand the temptation to remove the ability to vote down when you're eager to build an engaged audience. But I think if your goal is eventually to build a content-driven, high quality set of facts (whatever those "facts" may be: questions and answers, movie reviews, workout videos, or something else), you really have to watch out for broken windows emerging. For those who don't know what I mean by that, and who don't want to click the link, that's a blog post by Jeff Atwood from March 2009, in which he explains the importance of down-votes, and references Broken Windows Theory, which basically says that people who see bad content think it's acceptable, and thus you get more bad content.
But I've written most of this with a bias towards the task-based Stack Exchange model, and less-so towards the other side of that. That's because I just don't have a ton to say about the latter, not because I feel that it's worse, at least at what it's meant to do.
So, since I've already gone on for too long anyway, I think the one question you have to ask yourself is whether your site will serve people objectively or subjectively, and how much you want to moderate content purely on a binary (this belongs here, this doesn't belong here) level, or if you'd like to allow some more subjectivity. I wouldn't say one is better than the other, even in a small community, they're just applicable in different contexts.
Small communities are small. Therefore everybody knows everyone. People can grow closer due to this behaviour because the probability to interact with the same user more than once or twice is higher. Small communities are known to be more 'family-like', and opened to the community itself.
Do downvotes affect [...] a small community?
Yes, they most likely do. Since the community is small and users are well known, their behaviours begin to shine through, too. In a small community, you always know who will go against you, or downvote you. You always know who will be the one helping you out of the misery. In bigger communities, you usually don't know who downvotes you because of the size. There are more users, so there are more chances you get downvoted by several people.
Actually, it's nothing bad to know who downvotes you. But if you're about to build a community, it might be. The community splits if users are known to be down- or upvoters. The dowmvoters say that you need downvotes to keep within your scope, and to provide high quality answers. The upvoters say you need those question nevertheless since they are valuable to the community in generating traffic, or providing interaction.
I don't state that downvotes are bad but they won't really help you to grow fast. You will still grow even if you have downvotes, however, it takes much longer.
Imagine that you are an user who just discovered a community. You see that it is small, and that you can critically influence it. You start to participate, either by asking a question, or by starting a thread, and somehow you get downvoted or criticized. You ask why, and people answering you state something like 'Your content isn't within our scope. Please consider editing it so it is in our scope.'. What will you do next? Either you still are keen to participate, or you leave.
You can see something like this on SE sites all day. The bigger mass of users just leave then, some stay and actually edit their question(s) to fit the standards. The internet is fast so you can't expect that people edit every single question. That's the reason why downvotes hinder you but not damage you. You can grow with downvotes but it will take much longer to stay the goal audience.
In the end, it depends on your current situation. If you already have enough users that stand behind you even if you enable downvotes, you can go for it. You can easily improve the quality of your site then. If you are still small, you should consider it really carefully. There's no point activating it if you lose a critical mass of users expect you desperately want to improve your content. It doesn't matter which way you choose, but you should be aware that one way needs more time to build up a big community.