They have no specific dress code. You can wear anything casual.
The Cygnet Theatre is well known for its original theatrical plays and is similar but much smaller than the Old Globe however, I enjoyed the two plays that I saw there much better than the few that I have seen at other San Diego theaters. The ticket prices run about $25 and the theatre seating is quite comfortable; it doesn't really matter where you sit. I think that you get a better perspective if you sit towards the rear of this theatre. The cast members have a impressive repository of acting accomplishments. The appropriate dress for this theatre could be anything from proper casual to evening wear. Since the theatre is located in Old Town, I would imagine that they attract tourists who generally dress in comfortable casual clothes. On a positive note? At least you don't need "opera gasses," nor do you need to don "formal wear."
Like at most Southern California theatres, most people dress on the casual side here, including shorts and t-shirts. However, some people still choose to dress up.
Casual to evening wear
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