What makes you think illegals are getting minimum wage?
Anyway, the "minimum wage" is an artificial issue. Only about three percent of all hourly workers are working for the Federal minimum wage or less. About fifteen states have minimum wages higher than Federal, which if factored in would mean that the percentage of hourly workers working for local minimum wage or less must be higher, but not by much.
Block off the southern border, either by personnel or barriers or both. Once the inflow is dealt with, we can start worrying about those already here.