How can I catch that big long blue eel on the dinosaur island?

It took a long while, but eventually worked. You need a weight down to 790 metres ( I tried for ages with 690 and no luck) and then be lucky enough to hook it in the head. Any other part of the body doesn't work.
Pure luck, but got there in the end. Good luck.

I caught it on about the 5th try...You need to save your drill fuel, start out at 790m or 890m. Sometimes you can see it when you pass it on the way down. Catch a fish so that you don't go any deeper and watch the top of the screen carefully as your hook goes up. At the first sign of blue, push and hold the screen to activate the drill. If it was a megaladon then let it go and continue. Holding drill, hover around the long blue eel's (Basilosaurus's) head moving back and forth until it bites. You won't always get there with it's head accessible; if not you just need to re-cast and hope for the best. Make sure you don't hit a mine while you're celebrating on the way up.

I caught it using a 690 drill but it didn't intend to catch it. I saw it when I was going down then forgot about it but when I was coming back up I caught it when my hook caught it on the head

I used a 690m drill. Part of it is luck because you have to get it when it's head is exposed completely and even then you have to get it in just the right spot. Sometimes it takes a while.

Is it a Basilosaurus ? It's a big blue whale .

I caught it today using a 390 drill. It's all just about being lucky enough to get it's head

I caught it like three times today, using the 490 drill. If you get to the head, it's easy!