Hotel quoted in $, will that change with drop in €?

You will need to read the "fine print" to get a definitive answer to your question. That said, booking sites often quote in currency local to the person making the reservation as a point of information even if the contact is denominated in currency local to the hotel.

And if you pay using a credit card with conversion fees, your cost in North American dollars will be adjusted accordingly.

Some hotels offer discounted pricing with a no cancellation pay now policy. That would lock in your price if the Euro were to strengthen. And also lock in your price if it were to decline.

Check the contract terms. For example:, booking.com terms and conditions:

ttps://www.booking.com/content/terms.en-us.html?aid=311088&dcid=2&label=hotel-70114-at-cwCL2Qekpqin8r*9Trx9iQS24396519363%3Apl%3Ata%3Ap1%3Ap2%3Aac%3Aap1t2%3Aneg&sid=16d9e6c40b45a7cdcabe2c6574dc1265

The currency converter is for information purposes only and should not
be relied upon as accurate and real time; actual rates may vary.

I agree with Edgar. Most booking sites convert to "your" currency, allegedly in order to make it easier for you.
In doing so they are hiding the real price from you and muddying the waters. Your contract is almost certainly with the hotel, and in Euros. When you book it should tell you this amount (if not, do not book). The website is acting as an agent and collecting a commission.
If you pay the hotel with a card (debit or credit), then your card company will convert from Euros to USD using the rate on that date, so you may end up paying more or less than the rate the website quoted.

Just a heads-up. Don't fall for "dynamic currency conversion." That is an offer by an enterprise to charge your cc in your home currency. It's another of the myriad ways businesses, especially banks, have dreamt up to squeeze more money out of consumers. They'll give an exchange rate far less favorable than what the cc company will; and you'll probably still be charged a foreign transaction fee, if your card levies one. Insist on being charged in euros. And, btw, we've always been charged in the local currency, when using a booking site. They give you the US$ amount based on the rate when you book. Somewhere, probably in fine print, they say that and warn you'll be charged in euros, etc.

The hotel booking sites aren't hiding anything at all. At the top of the page you find all the currencies and languages you want. You can have the room prices shown in GBP, Euro, Dollars, Rand, whatever makes it easier for you personally to figure out if a room fits your budget or not. Since I live in Germany, they always give me the website in German and in Euro, but sometimes I switch it to dollars if people have posted that x amount of dollars is their budget.

You will pay whatever the rate is for that day unless you prepay the room. If you prepay, you can not then cancel.

I like using www.hrs.com to find rooms as their cancellation policy is very liberal.

Thank you all. I only prepaid my first night in so would have paid at that exchange rate. All my other rooms are reserved but yes, I did my search based on the $ just for ease of use so I think I'm all set.

A quote means nothing. Unless you prepay, it what you pay at the desk that counts and that should be always in local currency. I would not accept the hotel's conversion rate. You need to starting thinking in terms of Euro. And often there is a discount for cash payments. So ask.