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After completing the game on hard I would say the best way is to freeze her & then drill the hell out of her. Remembering to freeze her as soon as she gets loose, this along with armour piercing bullets for your Machine Gun will do the trick. For the earlier Big Sisters (before obtaining the Winter Blast Plasmid) Electro Bolt & the Drill works quite well but always try to maintain distance.

After the first Big Sister fight you have more control over where she will appear. Wait to save/harvest your Little Sister until you have found a place to fight her which is strategically advantageous to you. By that I mean a room with a Turret or Security Camera you can hack. A small room is good & bad. Good as it limits her acrobatic movement but bad because that means she spends more time in your face doing lots of melee damage.

Other helpful strategies include hacked Security Bots following you & deploying Mini-Turrets, both of these will keep her distracted while you unload on her. Good Plasmids include: Security Command levels 2 & 3 to summon Security Bots without the need of a camera. Hypnotize works very well if you have level 2 to make Splicers friendly, this means you can call up a small army to take her down. Hypnotize level 3 can be used to have a Big Daddy on your side in the fight.

If you use the Hypnotize Plasmid, be sure to switch off any Gene Tonics that have an area of effect when you take melee damage (Fire Storm, Ice Storm, Lightning Storm & Elemental Storm). If you are hit near your hypnotized buddy & the storm effect hits them they will immediately turn on you. Happy Hunting.