I am with BT and the Internet connection is very reliable !
In the last two and a half years I have lost Internet connection once, and that was for part of one day due to serious Local problem.
Don't know how reliable Virgin is, you will have to ask Virgin users about that. I am quite happy with BT at present.
AOG, the important thing is do you mean virgins cable service?
If so go with virgin, a cable connection is far faster and more stable than any ADSL broadband such as BTs service.
if you mean virgins ADSL broadband thats delivered over a BT line anyhow then it makes little odds who you pay your money to the actual connection will be the same.