Secure a principal role in a SAG production. You must be paid for this role, and you must have proof of employment. There is a huge Catch 22 with this method, though; under normal circumstances a casting director will not even consider you for audition on a SAG project unless you are already in SAG.
Receive three SAG "vouchers" from background acting (extras). When you work on a SAG TV show or movie as a background actor, you are given a voucher as proof of work. Your union affiliation is clearly marked on each voucher, so if you are nonunion you will receive nonunion vouchers. However, every SAG show that hires extras is required to hire a certain number of SAG members and the rest may be nonunion. In some instances, there is a tiny possibility that the show will have "spare" SAG vouchers which they can give to nonunion actors (if a SAG extra doesn't show up for the job, for example). Know that the show is not required to hand these vouchers out, though.
Production companies and background casting directors sometimes hand these vouchers out for other reasons, too. If you have a special skill or if the production company is in dire need of a particular type (I got one of mine because I had short brown hair), they may offer you a SAG voucher for the day. Also, if you are recalled to the set from one day to the next and there are fewer extras on the second day, you may be eligible for a voucher because the number of SAG extras has to remain the same.
If you work as a background actor full time, eventually you will get these precious vouchers. It is almost entirely random and there is not a lot you can do other than be professional and show your willingness and interest in an appropriate manner.
SAG has a special exception for people who work in principal roles under the jurisdiction of one of their affiliated unions: the Alliance of Canadian Cinema, Television and Radio Artists (ACTRA), the Actors' Equity Association (AEA), the American Federation of Television and Radio Artists (AFTRA), the American Guild of Musical Artists (AGMA) and the American Guild of Variety Artists (AGVA). If you are a member of one of these unions for at least one year and you get a principal role in a production under that union's jurisdiction, you are eligible to join SAG.
Not everyone should join SAG. Remember that once you join, you may not do any more nonunion work so if you are just starting your career and you are still trying to gain experience, you may be better off working nonunion for a while to build up your resume and your reel first.