I think the best approach for this is thinking about the search space and the fitness landscape. If different components of the solution vector can independently improve the fitness crossover operators will have a positive impact.

Another aspect is the search space itself. Is it real-valued, is it binary, is it a tree-like structure,..?

Traditionally genetic algorithms are operating on binary encodings, and they often work ok problem which have binary solutions (a fixed-size vector of bits). These problem do not have gradient to start with. However one should investigate beforehand if there are combinatorial approaches to solve the problem.

For real-valued problems with no gradient: evolution strategies with a smart mutation operation like CMA (covariance matrix adaption) would be a good choice.