This is one of those things that I'm planning to work on more actively soon!
I think it can be an effective way to help people find Lemmy, and it will work better if that the focus is on letting people know that the option exists (rather than being pushy about it).
Which community to work with
IMO it's the best community to work on might be one that you are already familiar or active with. If you have a good reputation with the mods already, they will be more likely to listen to your idea. Similarly, some subreddits will benefit from having a space on Lemmy more than others.
What benefits to highlight
I've been busy with some other things for some time, but when I was last working on this with a few subreddits, the main benefits were:
A lot of users don't want to have to use Reddit, and many have already stopped using Reddit. Having a similar community without ads/tracking with the same moderation team will go a long way to helping those people stay involved
A well known backup community is very helpful. Some subreddits serve important purposes and even temporary outages on Reddit can affect time sensitive questions. Meanwhile, there have been cases where entire subreddits were removed in error (ex. automated systems reacting to mass reporting), and users had no idea what happened. It's helpful to have a second place that everyone knows to check for updates in those situations.
Potential process
bring up the idea, and offer to help with the set up and day to day moderation
work with the current mods to make moderation consistent between the platforms, so that the experience is similar
see if some of the mods want to make accounts here so that they can rest easy knowing they have moderation permissions here, even if they don't use it day to day
Once both communities are linked, there's a chance for trust to build and users will feel more comfortable knowing that the Fediverse community is an 'official' one. That can be helped by having information in the sidebar, a pinned post, and/or a note on the submission page, etc.
Day to Day
This requires having an account on Reddit and being somewhat active on it. Users are more likely to explore the fediverse if they see content from it. Some ideas could be
A weekly "here's what you missed from Lemmy"
Sharing any major updates about Lemmy / the fediverse community
If there is an important post that could benefit from reaching more people, then letting the OP know that they can post here as well