[Answer has been edited to incorporate comments.]
asikarazu is the negative form of the adjective asii, not a verb. That being the case, adjectives acquire much of their conjugation via suffixation of the verb ar-. This is asiku + arazu, and the u drops. arazu is the verb ar- in irrealis + negative -zu.
Another example of this construction would be 必ず ie 仮 + ならず which also fits this 形容動詞 + ならず pattern
[Bonus question: What is the kanji and literal meaning of すみません？]
I answered this recently. Please check the comments.
(Note: When used to apologize, sum-i-mas-e-n means "(I have done something rude to you, and my feelings) are not at rest". When used to be thankful, the sense is "(I am unable to return the favor, and my feelings) are not at rest". It was not my intention to say that all sumimasen is a contraction for ki ga sumanai / sumimasen. Phonologically that is not very likely. Rather, grammatically, a single adjective or verb can be a complete sentence. Many things are not said when unneeded. As such, I was filling in the missing blanks to assist with compression. If it helps to interpret asikarazu, think of it as 気を悪くしないで or 悪く思わないで. )