If I were Irish I'd be hopping mad at the comparison. The Irish didn't violate our laws to come here, didn't come in the expectation that everyone would speak Gaelic to serve them, didn't want to annex parts of the USA to Ireland, didn't fill our prisons with wildly disproportionate numbers... in short, they didn't come here to turn the USA into another Ireland, they came to become Americans.
Just as a fact, I am quite confident that a count of prisoners c. 1859-1900 would have found "wildly disproportionate numbers" of Irish, as compared to native stock, in the big cities of the East. The truth is that almost all immigrant groups are disproprtionately represented at the bottom, and that includes crime, until they get established. That fact doesn't determine the immigration debate either way, but it's just not correct to deny it, and imagine a golden past when all immigrants came over and immediately turned into maodel citizens. Most did, but it took time, then or now.