First, make sure no voidwyrms are nearby, as they will try to kill you. Next, pet the Astrodelphis. Finally, put element in the last slot of your hot bar and start feeding it to the Astrodelphis. Thanks for reading my tip!
U have to pet it and feed element. The amount of element goes up each time u have to feed, but the taming bar will fill up faster. Easy element u can gather by using an anky or pickaxe on the blue crystals in the space between the bridges. In the center of the map
Swim up to the dolphin until it notices you. If it squeaks, you're in business, if you see red sparks from it's blowhole and it screams, you're in trouble. Keep hard element in your final hotbar slot. Dolphins aggro every other dolphin when fed. Make sure the one you want is alone. Always keep the dolphin between you and the direction you want it to go, because it will run away occasionally and you don't want it running into a pack of other dolphins or a wyrm. Sometimes they swim away after 3 pets, 2 pets and a feed, or just one pet. The bite during taming doesn't hurt much so just stay away from the mouth. The bite during true aggro hurts a lot.
Biggest tip: when the dolphin swims away STAY RIGHT ON IT because the taming bar will absolutely plummet 5-10% every second if you don't pet it fast enough. Because of this bring way more element than you'd expect to use. Around 3 stacks of 100 each should be enough. You never know when you'll have to reposition your target or avoid hostiles so be prepared to lose some progress.
BEWARE OF FOOD DROPPING. Not sure if it is a glitch but within a minute or so of taming it was starving. Filled with with a few stacks of meat and it ate them all and starving within another few minutes. Bring extra meat.
Build an argy trap in the space bridge. You know, 4-5 gate frames in a row with gates at each end, like a tunnel. With enough space between for you to run. Fly up to it and let it “aggro” on you (give you a friendly nibble) then fly to the trap and let of follow you inside. Close the gate. Soooo much easier to tame this way. No more flying off between bites.