I've been reading Alexander the Great: Lessons in strategy by David Lonsdale (if there's any major criticisms of the book I'm interested to hear those too), and he talks about how Alexander attempted to circumvent or neutralize the dominance of the Persian navy by capturing the ports in Anatolia and Syria that were required to keep the Persian navy operational.
However he doesn't address how Alexander actually managed to get his army across the Hellespont in the first place. It certainly seems as if a superior navy could prevent anyone from crossing the strait.
The Persian navy was large and powerful, even more so than the combined fleets of the Greek city-states under Alexander's control, however even a smallish sea like the Agean is still vast, and the Persian fleets couldn't be everywhere at once. Besides, at the time Alexander made his crossing he had not attacked any Persian territories yet, so the Greek mainland wasn't under any sort of naval blockade by the Persians. Sailing across the Hellespont and landing in Persian-controlled Asia constituted Alexander's "declaration of war." Even if the Persians were more or less expecting him to make such a move sooner or later, he still managed to achieve "tactical surprise" for long enough to get his army across the Hellespont. By the time the local Persian authorities had gotten an army together to oppose him (at the Battle of the Granicus) Alexander had been in Asia for many days.
According to Arrian his invading army was about 30k soldiers strong, "and they were transported in 160 triremes, besides many trading vessels." The ships would have come mostly from the various Greek city-states Alexander had spent the previous few years re-conquering.
Alexanders Father Philip had already established a presence in Anatolia prior to his death. Having a land army on both sides would have prevented any Persian fleet from being able to put put in a night and collect water and cook meals. This in addition to the fact that the crossing from Europe to Asia is very narrow and a small number of ships could achieve enough local superiority to enable a crossing in spite of a much larger fleet. The real question is why the Satraps didn't attempt eject the Macedonians prior to Alexanders crossing.