What kinds of internships would be good for someone seeking a career in theatre?

There are in fact many internships and apprenticeships in acting all around the country. If that is your dream then you should definitely pursue that. There are even some that will give you the opportunity to earn points towards joining Equity.

If you are wishing to broaden your horizons though there are also many internships/apprenticeships in stage management, props, carpentry, sound, electrics, wardrobe, costumes, etc.
You can always check sites such as Offstage Jobs, Backstage, Art Search, Playbill, StageJobs Pro, or even just simple Google searches to find opportunities.

Best of Luck!

Take an intern position as the stage hand. While you learn a lot of technical skill, which is useful, you'll meet other actors and can learn from them. It's a side door approach but you would be surprised what you can learn as a fly on the wall. Never quit, give up, get discouraged, be polite, pleasant and professional. The doors of opportunity will present itself.

Seek internships at Regional Theaters, Television and Film Studios. Seek to broaden your skills ... and learn from observation. Best of luck.