Your translation is indeed correct as such, but might be misinterpreted, mainly because sens is written and pronounced the same way singular or plural. Out of any context, beaucoup de sens might mean a lot of sense.

I would indeed use plusieurs significations, or différentes significations but if you want to use your locution, I would then say:

… beaucoup de sens différents.

The différents here implies plural form of sens, translating to meanings instead of meaning.

You could also say: « X a de nombreux sens », « X a des sens multiples » or « X a bien des sens ». It is difficult to translate an expression without knowing the rest of the sentence.