This seems to be an idea very far removed from our current knowledge level about physics to even begin to speculate. This assumes that we somehow would have the ability to create new universes and transfer entropy into them. Those are massive assumptions. Given those assumptions it isn't at all clear that the idea would work. The specific physics would matter. For example, if there's a maximum amount of entropy that can be somehow transfered into any given universe and creating new universes takes lots of entropy then this won't work. Ask again after we have a Theory of Everything or at least a better idea of how to reconcile quantum mechanics with general relativity.
Could you also chuck away all the cold into these basement universes?
Entropy is like phlogiston. It's not really the object in question, negentropy is.
If the creation of basement universes proves to be useful in producing negentropy, or energy, without obedience of the laws of thermodynamics, then that's awesome. But you'll need to provide a reason why I should expect:
A) the creation of basement universes to be possible and B) It to be possible to extract negentropy and/or energy from these basement universes.
There are bounds on the entropy that can be contained in any given region, but I'm not sure whether the reason we believe these bounds to hold implicitly assumes no basement universes exist. Does anyone know?
In the linked comments, Mitchell Porter points out you'd need an infinite amount of stuff to throw into the basement universe, though for all I know whatever technology allowed you to create basement universes in the first place would let you solve this also.
Wouldn't the Second Law of Thermodynamics mean that transferring entropy this way would, in turn, generate entropy in its own right? You might be able to make the universe last longer, but I don't think you'd be able to make it last forever. Even if you could, though, you'd still run into the problem of proton decay eventually.