The people behind Monzo and Starling seem legitimate enough; Anne Boden was COO of Allied Irish before founding Starling. Tom Blomfield founded Go Cardless, then became CTO at Starling before founding Monzo. Both have a UK banking license, so up to £85k is covered under FSCS. The only reservation I have is profitability. Both currently make substantial losses, which is to be expected. But the path to profit relies on people buying in to the idea of one account for everything. Both companies want to manage all of your money for you. Mortgage, loans, savings, insurance, investments, all from a range of third party providers. Only time will tell if this is something people actually want.
For now, neither is likely to go bust as VCs seem quite content to keep taking a punt on it all working out.