"Busy" is an adjective, so it can only describe a noun, not a verb. Since "day" is a noun, "My day was busy" is correct. "My day went busy" is incorrect because busy would be describing how the day went, not how the day itself was. Busy cannot describe the verb "went" because it's an adjective. But you could use the adverb "busily" instead. An adverb describes a verb, so you could technically say "my day went busily." But this would be a strange usage, so just stick with "my day was busy".
Both "how did your day go" and "how was your day" are correct.