there is perhaps some historical connection between the く sound and い sound, either phonologically or semantically.
I think the answer from blutorange addresses this.
Maybe these two classes of words [〜い adjectives and 〜く verbs] diverged from the same class of words somehow?
I'll disagree with blutorange about this part, as his answer is (I believe) ...
Yes, for example consider the beginning of 枕草子【まくらのそうし】:
If you rewrite the adjective and verb forms to their modern form:
Observe that an adjective such as 美しい was originally 美しき (before a noun, 美し at the end of a sentence), with a shift of き→い, see イ音便. 咲いて was originally 咲きて, where you can ...