However -- this is actually the easier phrasing to see it in. If it were the other way around, "A variety of activities is available", it would be much more of a natural reaction for people to say "a variety of activities are available" because it sounds like activities are available rather than a variety is available. And that's one where you might just go with what sounds natural, as variety is a collective noun. It's easier to see it as "there is a variety" because "a variety" sounds more singular.
But if it's dialogue, or someone is making a sign or something, it should be whatever they would say, regardless.