Companies paying dividends usually have a solid business plan which allows them to do so even during recessions
This will be company-dependent. Some companies may experience a cash crunch where it is in their best interest to suspend dividends.
Therefore the distributions will continue during a crisis
What I would be more worried about in a crash is the value of the ETFs. Your dividends may go down, but the yield (as a percentage of the value) may actually increase because the value of the ETFs will (presumably) drop.
As an example, look at VIG in the second half of 2008. The fund paid three quarterly dividends - 0.281 before the September market crash, and 0.268 and 0.257 afterwards. Yet the value of the fund dropped by almost 25% even accounting for the dividends. So your income was relatively stable but your growth potential was significantly reduced.
Also note that dividends from an ETF are offset by a drop in value of the ETF, so from a wealth standpoint, it is a wash. It's mechanically the same as if you owned an ETF that paid no dividends and sold, say, 2% of your balance every quarter.
If you want low-risk investments with more stable income, then you might look at fixed income ETFs. They will be less susceptible to drops during a crash, but also won't grow as much in bull markets.