OP YANBU to feel they shouldn't have had sex in your home but you were perhaps naive not to consider that they might.
Also I think it's unfair that you are getting all these negative comments about the cameras.
I grew up living on the premises where my parents work and got used to having cameras, locks and alarms all over the place.
We/they often get visits from both the police and neighbours on the street if something has happened, asking if they can check the cameras to see if they picked anything up.
We don't have cameras on/in our house or garden now but every so often I do consider getting one for the front porch and one for the parking area at the back of the garden. My friend has a system that works through her TV and if someone knocks at her door she just changes the channel to see who it is before she answers.
As someone who is frequently stalked by my IL's I would love that system, I just can't explain why to DH 
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I don't think you are voyeurs to have some home security or to check it when asked, and lets face it, if one of the cameras is picking up footage inside the house it's you and your family that are mostly being filmed.
It's also unreasonable to think you still have the film or assume you were using it to check up on the babysitter.
And nowhere did you say that the problem you have with this is that your children were in the house. You actually said they were upstairs asleep and so no complaints there.
They were there to take care of the children but that doesn't give them the free run of your entire house.
Living room, kitchen, bathroom and kids bedrooms (and whatever corridors they need to use to get there) are fair enough but the rest of it should be off limits.
If they hadn't wandered out of the areas you rightly expected them to stay in then they wouldn't have been caught on the film, as you say it's at the other side of the house.
I can also believe they were stripping off in a corridor. If they are young enough to have sex whenever and wherever they get the chance why wouldn't they be stripping off and leaving a trail of clothes through the house on the way to the bedroom (or the dining room table, wherever they did it?). We've all done that haven't we? 
I might feel a bit violated if someone I trusted was roaming around my house when they were there in a position of trust, with or without the sex.
Although I would actually expect babysitting teenages to get up to something, with or without permission for the boyfriend to visit too (which is why my fifteen year old neice is never asked to babysit LO).