整 should be used after the measure word, so
三百八整人 is not correct.
三百八人整 is weird too.
整 is usually used at the end of the phrase to prevent from adding new numbers to change the whole meaning, especially in financial affairs. Such as
三元 -> you can change it to
三元整 -> you can't change it to
三元整五角八分, which is wrong expression.
三百八人 can't be changed to
三百八人零点五人 and so on, so need not to add
整 after that.
BTW1: It also can be written as
正 in financial affairs.
BTW2: As @user58955 said,
三百零八人 is more natural than