Re: Do your babysitters allow TV?
Babysitters - no, never. I specifically hire teens who will play with my kids. The TV stays off until the kids are asleep.
Daycare - yes, sometimes. When watching my and other children full-time, I understand that sometimes there might be a "need" for some TV time. I use TV at home in order to do DD's hair. Otherwise it is off. While I can cook/etc w/DD in the room, I recognize that others might not, so as long as it's not on ALL day, I'm okay with a little TV time at daycare as long as the shows are appropriate and commercial-free. If daycare has the TV on all day, I find a new daycare.
If this babysitter is only watching your child for a few hours AND only has your child, I feel it is perfectly appropriate to ask her to keep the TV off and play indoors/outdoors with your child instead. Be nice when you talk to your babysitter, but remember that she is your employee, not your friend. She needs to be doing what you expect of her. Make sure you listen as well. She might have a valid reason for the TV time (ie she's putting away your sharp knives, then sits with your child until the end of the show). Listen, talk, listen, and find a solution that works for both of you.
S, mama to my First Grader, K (9/07) and my PreSchooler, e (7/09) and married to my best friend, E (2007)