Here are some strategies that I have found helpful at Mi Yodeya, whose community started as a SE 1.0 community that I founded:
Lead by example
Start by demonstrating the best example you can of active and responsible community participation and stewardship, so that other active users can get a good sense of what this looks like and can emulate it, if so inclined.
When you notice users stepping up to take care of meta-tasks, explicitly thank them through comments or in chat. Some examples of when such praise might be particularly appropriate/effective:
When someone executes a particularly useful and labor-intensive edit.
When someone leaves a particuarly sensitive and helpful comment.
When someone does some good work in cleaning up tags.
When someone earns any of the "Moderation Badges" that are awarded for doing things like voting and editing. The bronze badges are generally cases of a new user doing a certain meta-activity for the first time, which is worthy of encouragement. The silver and gold badges are cases of sustained involvement, worthy of congratulations and gratitude.
Showing such appreciation provides positive reinforcement to its targets, helping them feel good about their contributions and be more inclined to keep participating. In addition, it sends a signal to other community members that such participation is valuable.
Ask for help and advice
If there are outstanding questions of site policy to be resolved or cleanup tasks to be performed, ask explicitly for help in your Meta site and chat room. If there are particular users who you think may be interested in helping if prodded, prod them in chat or by comments on related posts. You may get help from people who didn't otherwise know that their help was needed or wanted, resulting in their becoming more active in such efforts in general. In addition, asking people specifically for their opinion could make them feel like you value them, making them more likely to want to contribute.
Welcome and guide new users
When someone posts their first question or answer, post a comment explicitly welcoming them to your community and providing pointers to meta-content that might help them better fit their contributions with community expectations and also to other site content that they might find interesting. Most of these comments, posted to contributions from drive-by users, may never get read by their targets. However, some will not only get read but will be deeply appreciated, and may make the difference of converting a drive-by user into a community member.
You can make this task easier using pro-forma comments. Posting similar "Welcome ... thanks ... please take a look at ... you might be interested." comments to many different posts may feel boring to you, but for each new user, this is likely the first time they encounter that greeting, so if it's written to sound warm and welcoming, there's a good chance they'll receive it that way.1
Invite people to Meta and chat
When someone posts content or a comment that gets at interesting issues of community standards, point them at a relevant post on your Meta site, invite them to post a Meta question, or invite them to come to your chat room to discuss the issue. Again, most drive-by users won't bite, but the few who do could be drawn into more active engagement with your community.
Keep things fun in your chat room
Try to make your chat room a place where, among other things, there's frequent fun banter related, at least tangentially, to the topic of your site. A fun atmosphere in the chat room provides a place for people to engage with the community outside the strictures of the SE Q&A model, so people who can do valuable Q&A but also enjoy a good free-form chat have an outlet, within your community, for the latter. You may also end up with users who mostly come back for the fun chat, but who can also be led to participate in your site's main and meta content from time to time when it comes up in chat.
1. When cashiers at Walgreens tell people, by rote, to "be well," it rubs some people the wrong way, while others (including me) consider it a "nice touch" of humanity and appreciate it each time. In any case, what makes it obvious that it's by rote are the customer's repeated encounters and the tone of voice of the cashier, both of which are absent when you post a comment to a new user.