Will a heavier umbrella base stop my umbrella blowing over in a storm?

Only up to a point, and the limit is the umbrella rather than the base. Our wind-optimised centre pole umbrella is rated to Beaufort 4, about 25 km/h. That is the canopy's own limit, and no base raises it.

What the base does is hold the umbrella down up to that rating and stop it tipping or walking in a gust below it. Above the rated wind the correct action is to close the canopy, because a closed canopy presents a fraction of the open sail area.

So on an exposed site the answer is two things together: specify the heavier base and close the canopies at the rated wind, or specify a permanent in-ground position so the load goes into the ground. A heavier base on its own is not a storm solution.