How can I stop water from coming in through a crack near the top of my bulkhead wall?
Best bet here (given that it's shallow) would be [in the spring/summer] to dig up around the outside, and waterproof it from there. Waterproofing the inside only is doomed to fail, as the water will still be in the wall, and in this arrangement it will freeze and move, breaking any inside-only treatment. Dig down a foot or so, clean the wall well, patch any holes or cracks, and then coat the whole thing with waterproofing.

Your stairs are almost an exact match for the circa 1964 house I worked on for a while ;-) A center runner was evidently not popular then.