It is thought to have derived from Welsh and is often considered derogatory. Use renege or other wording instead.

Online Etymology Dictionary

Etymonline.com says of welch:

1857, racing slang, "to refuse or avoid payment of money laid as a bet," probably a disparaging use of the national name Welsh.

And of Welsh:

Among the English, Welsh was used disparagingly of inferior or substitute things, hence Welsh rabbit (1725), also perverted by folk-etymology as Welsh rarebit (1785).

Oxford English Dictionary

The OED says of the verb welsh or welch:

Origin uncertain; perhaps < Welsh adj., on account of alleged dishonesty of Welsh people (see note). Earlier currency is probably implied by welsher n.1, welshing n., and welshing adj. Sometimes considered offensive in view of the conjectured connection with Welsh people.

Their first quotation meaning to renege on a betting debt is from an 1860 Racing Times:

The plaintiff denied that he had ever..‘welched’ a man named Williams at Worcester in 1854.

Their first quotation of noun welsher, a bookmaker who refuses to pay, is from an 1852 Racing Times:

One of the above fraternity [sc. betting impostors] was observed following his calling, by a former victim... The ‘Welsher’ sneaked off to another corner of the field.

Their first for noun welshing is from an 1854 Era:

The subterfuge and welching of the betting enclosure.

BBC

But it is still used, often by politicians, including the BBC itself. Occasionally they apologise. The BBC reported in February 2012 that Education secretary Michael Gove apologised for saying he'd "welshed on the deal" in the House of Commons, and 'Bill Clinton apologised to Republicans in 1995 for calling them "Welshers"'.