If the equity market in the USA crashed, its very likely equity markets everywhere else would crash. The USA has a high number of the world's largest businesses and there are correlations between equity markets.
So you need to think of equities as a global asset class, not regional.
Your question is then a question about the correlation between equity markets and currency markets.
Here's a guess: If equity markets crashed, you would see a lot of panic selling of stocks denominated in many currencies, but probably the most in USD, due to the large number of the world's largest businesses trading on US stock exchanges. Therefore, when the rest of the world sells US equities they receive cash USD, which they might sell for their local currency. That selling pressure would cause USD to fall. But, when equity markets crash there's a move to safety of the bond markets. The world's largest bond markets are denominated in which currency? Probably USD. So those who receive USD for their equities are going to spend that USD on bonds. In which case there is probably no correlation between equity markets and currency markets at all.
A quick google search shows this kind of thing