Let us first understand what is liquidity. In the traditional sense of a bank, a liquidity crisis occurs when most of its funds are locked up in long term assets and a lot of depositors ask for their money. Asset classes like real estate and private equity may also be considered illiquid, due to time and effort taken to find buyers.
As far as crypto exchanges are concerned , they are just supposed to hold your coins. In case there is a shortage of buyers at an exchange, you always have the option of withdrawing your coins and selling them at another exchange. So if an exchange has problems, it is not going to be about liquidity.
The price goes down because more people are selling than are buying. This means that more coins are available for purchase. I suspect that the OP is postulating that because of the drop, then exchanges are less likely to want to increase their inventory, and thus the coins will be harder to sell through exchanges. However, there is always the ability to use your coins to purchase assets, and therefore avoid the extra costs involved in using the exchanges. In fact this can be a good chance to profit from arbitrage.
