Merriam-Webster's Eleventh Collegiate Dictionary (2003) identifies the relative vulgarity of various meanings of ass for your convenience. Here is how the various definitions rate:
"any of of several hardy, gregarious or Asian perissodactyl mammals (genus Equus) smaller than the horse and having long ears; esp : an African mammal (E. asinus) that is the ancestor of the donkey" MW VERDICT: degree of vulgarity not discussed (= not vulgar)
"a stupid obstinate, or perverse person {made an ass of himself}—often compounded with a preceding adjective {don't be a smart-ass}" MW VERDICT: "sometimes vulgar"
"BUTTOCKS—often used in emphatic reference to a a specific person {get your ass over here} {saved my ass}" MW VERDICT: "often vulgar"
"ANUS" MW VERDICT: "often vulgar"
"SEXUAL INTERCOURSE" MW VERDICT: "usu. vulgar"
"used as a postpositive intensive esp with words of derogatory implication {fancy-ass}" MW VERDICT: "often vulgar"
You could walk through these various definitions with your child and clarify the conditions under which Merriam-Webster deems ass to be not vulgar, sometimes vulgar, often vulgar, or usu. vulgar—or you could simply issue a blanket approval or condemnation of the word.
My sense is that views on the putative offensiveness of ass vary tremendously in different communities and subgroups of people in the United States, so Merriam-Webster's assessments of the various definitions' vulgarity are unlikely to accurately reflect community standards in an arbitrarily chosen locality. On the other hand, if you are at all concerned that people might overhear your child using the word ass and consider him or her vulgar, MW's appraisals suggest that blanket condemnation is probably the least potentially embarrassing position to adopt.
I doubt that the status of ass as a vulgarity (or not) has changed substantially in the 12 years since the Eleventh Collegiate Dictionary appeared, although people who watch network and cable TV probably have far greater insight into this question than I do.