about 7-10 minutes to the end closest to the port. Right across from LaQuinta hotel. There are plenty of taxis at the port - we were there first week of June.
But the Eagle beach then continues farther out all the way to the Butterfly farm (Divi Phoenix hotel area) - also nice to see. That's what we did back in 2016 - Butterfly farm first (not a ship tour - just took a taxi) and then crossed the road to the beach. Took a taxi back later.
This time we were going to go back to the ship, change and go to the zoo (Phillip's garden), but it was VERY hot and windy that day, and we don't like hot all that much, unless we are in the water. So, we just went back to the ship instead and stayed put. Definitely use a lot of SPF - we burned pretty badly, despite all the reapplying.
If you go to the Eagle beach not in front of a hotel, get there early if you need an umbrella ($40 for one with two chaises). You can choose to tell the taxi driver "Eagle beach" or Eagle beach in front of "insert high or low rise hotel name" which is $1 more.