When fighting a Big Daddy, use armor piercing rounds from the machine gun or pistol. For a more effective way of killing, launch frag grenades and run out a door. The Big Daddy will follow you. Fire proximity mines at the door, and when it walks through it will be damaged or destroyed.
Once you have obtained the crossbow and at least five Trap Bolts, it is easy to kill a Big Daddy. Face a wall and stand about three feet away. Make sure it is not something that the bolts can bounce off of, and you have at least a ten foot long straight wall. Once you are better skilled at this technique, you can try for shorter lengths if desired. Point straight ahead and fire a bolt. Take about two steps to the left or right, depending on your location and which way the Bid Daddy is facing, and fire another. Once you fire all five bolts, shoot the Big Daddy once with anything. If done correctly, it will run through the line of traps, taking enough of his health away so he can be killed with just a few more shots. Note: This may not work in an open area because Big Daddies strafe frequently. If you have the damage increase for the crossbow, it will only take three or four bolts to kill him. Do not crouch while setting up the traps. If the Big Daddy walks another way and gets out of sight, you may need to go back to get him. If you do get him, you can crouch under the set traps. An easy way to trigger bolt traps is to pick up something with Telekinesis and drag the object through it.
Once you get the Security Bullseye Plasmid, set off an alarm either by hacking or being spotted by a security camera. Then, throw the Security Bullseye at the Big Daddy. You may have to do this more than once depending on how much health the Big Daddy has when starting out. Try to avoid the battle because the crossfire and exploding bots will damage your health. The only problem with this technique is that once the Big Daddy is taken down, the bots will come after you. Be prepared to either run and hide, fight them off, or pay to shut them down.