To further bomb’s point — although living in NY means that as a Republican you won’t affect large elections, on a local level your vote can make a difference, and it is often the local level politicians who are very important to local mosdos and even private individuals. These local politicians are very in tune with the local demographics, and keep close tabs on who votes, often with staffers monitoring on voting day who actually comes out. If the numbers show that our community is not likely to come out, they will be that much less likely to help out the next time a local institution needs help.
As far as jury duty goes, I’ve been registered to vote in NY for 11 years and have not once been called for jury duty, and I have friends who are not registered who have been called more than once in recent years. Jury lists are taken from numerous sources including driving licensce lists, property owners, etc.