Yes absolutely. The depth of a rim is directly related to the side-area, and a gust of wind will have more of an effect on a larger surface.
The largest wheel surface area is a solid disk, and even pros will prefer an open wheel to a disk wheel on a race day with a gusty wind forecast.
If there are a lot of sidewinds anticipated, an old-school box-section rim presents the smallest side-area to a wind, and will react the least to a sudden gust.
Consider also that a front wheel has two pivot points - the head tube and the ground contact patch. The wheel's axle is not relevant to steering. So there is more of the wheel/rim in front of a line between those two pivots, and a side wind will push the front of the wheel more than the rear of the wheel.
The result is a net turn away from the wind. If the rider is not balancing and ready to react instantly, they'll find the bike turning under them without the rider's mass following, leading to a loss of control, a wiggle/swerve, and potentially a fall. Not good in a pack!
The only correct action is to prevent the turn from starting. You have to hold firmly onto the bars, and react immediately if there's a gust. If you're watching the grass on the roadside, or other riders ahead, you can see a gust-front coming and brace yourself for it. With timing and luck, you can even turn into the wind as it hits you, resulting in a net-zero steering change.
The other good choice is to slow down in gusty areas. If your speed was halved, you'd have twice as long to react to the gust before overbalancing. (very approximately).
Additionally, don't ride too close to obstructions, the edge of the seal, other riders, or cliff edges. Having some more run-out room helps recover.
Deep rims on the back have less of an effect because they don't get involved with steering. A solid side wind can push the bike around, but its not going to steer your bike into a co-rider or a parked vehicle, or off the road completely.