I think it's a great question and one that troubles me a lot, but easy answers are hard to come by. As Baptiste asks, what is a EUR investment anyway?
Fixed interest (bonds and other forms of debt) is simple enough, at least in principle: match your predicted future liabilities with debt instruments in the same currency and with the same maturity dates. That's what insurers do. If you think you'll need 100k in EUR in 2034, then buy a ten year EUR bond. You're sorted, aside from the bothersome little details like inflation (maybe by 2034 you'll need 120k) and opportunity cost (those bonds yield pitifully little).
Equity is even harder. A share that's denominated in EUR might buy you rights to a company that earns most of its profits in the USA or China and will perform in line with those countries and their currencies. Swiss Re, traded primarily on the SIX in CHF, has switched to reporting and paying the dividend in USD, because the USA is so important to the business. I own other companies that are traded in New York and quoted in USD, but do almost no business in the USA. I hold BASF, quoted in EUR, but it's very much a global business. My head hurts.
Short term, it appears to me that interest rate movements (or expectations) cause share prices to move in inverse relation to exchange rates and everything USD/EUR balances out, more or less. I can therefore sleep at night, having a lot invested in USD and most expenses in EUR. The risk is that there is some major dislocation (e.g. rapid unwind of the JPY/USD carry trade, death of the petrodollar) that causes USD to drop off a cliff. I guess in that case I would move my main expenditure somewhere else.