A discrete group has the property you describe if and only if it is virtually abelian. I must admit to never learning the proof, but some foraging on MathSciNet indicates the result is due to Isaacs and Passman:

Isaacs, I. M.; Passman, D. S.
Groups with representations of bounded degree.
Canad. J. Math. 16 1964 299--309 |
MathReview

(I seem to remember hearing that there are proofs which get better estimates on the degree of the abelian subgroup, but I may be misremembering.)

The class of virtually abelian groups has good hereditary properties. However, it is not stable under semi-direct products: indeed there are metabelian groups, arising as crossed products, which are not virtually abelian, e.g. integer Heisenberg, or the ax+b group over the rationals. Indeed, in the former case once can write down infinite-dimensional unitary irreps. So I think you need to restrict the kinds of extension you are considering if you want this property to be preserved.

If you are interested in the problem beyond the discrete setting, then I think one place to start would be with the paper of C. C. Moore

Calvin C. Moore,
*Groups with finite dimensional irreducible representations.*
Trans. Amer. Math. Soc. 166 (1972), 401--410. |
MathReview

(A locally compact group $G$ is called a Moore group if all its (continuous) unitary irreps are finite-dimensional. It is a hard result of Thoma that if a discrete group is Moore then there is a uniform bound on the degrees of the irreps, which is the property described in your question. Of course non-discrete compact groups show that the two concepts are in general distinct.)