Gonna be a lengthy one... I hope I am able to convey my thinking with some clarity.
Imagine that cash is precious and that, apart from all the "normal" risks you face when lending it to someone (e.g. counterparty risk), there is also the possibility you wouldn't have enough on hand to deal with any of your own issues that may arise. This fundamental principle, IMHO, is the underlying reason for the existence of all the different bases out there (tenor basis, x-ccy basis, etc). Since nobody can model or quantify liquidity, there is simply no good way in finance to deal with these effects, which is why they always come out as "fudge factors" that make all the traditional "no arbitrage" logic work.
X-ccy basis is a specific manifestation of this effect, where there is a perception that USD cash is the most precious cash of all. Lending out USD cash makes you feel the most vulnerable and exposed, which means that you'd need to be paid extra to do it.
Obviously, this, IMHO, is sort of the underlying basic principle. In reality, many things may affect the way you feel about lending out your precious dollars. For instance, if you were of the opinion that the supply of USD relative to all the other stuff will decrease in the future, you might feel more reluctant to lend your USD. Likewise, if you felt that you (and all your fellow mkt participants) are likely to have more demand for USD in the future (maybe due to increased regulation or fear of a genuine crisis), you'd also be less willing to part with your USD cash.
Pre 2008 all such effects, while always present in theory, were happily ignored, because in a world of infinite banking system liquidity, the extra idiosyncratic value of USD cash was 0. So all these "arbs" were never really "arbs", but we were happy to pretend they were. Not any more.