Don't let her be on your hip all day.
A baby this old should be used to playing in a playpen or a gated baby-proof room where she can see you at least most of the time. If your parenting style keeps your child on you during most of her waking hours, separation is scary because it is new and different. You can't expect a child to sleep alone at night when she hasn't yet learned to play in a playpen while you are out of eyesight for a minute or three.
Start by putting her down with some interesting toys while you do a common chore (folding laundry, washing and putting away dishes, putting away the groceries, etc.) Make sure she has line of sight on you 90% of the time, and that when she doesn't you are still in earshot so she can hear you. If she's totally unused to this, you may have to start by making those "disappearances" super-short -- 10 seconds, then 20, then 30, then a minute, then two... you get the idea. If she gets uncomfortable while you are out of sight and starts crying, just call to her from the next room to let her know you are still around, but give her no other special attention for crying.
Start with a good (and by "good" I mean "unfailingly consistent") bedtime routine. Brush teeth, put on jammies, get snuggled into bed, read a story. Do it at the same time in the same way every night so that it becomes familiar and comforting.
Once the bedtime routine is over, go to some corner of the room and read a book, or use a laptop (on very low light with the screen turned away from her). The key is to be quietly there but not to interact with her in any way. She will cry and protest, but eventually she'll get the hang of it. The more she's used to having you at her beck and call the more she will have a fit -- you've taught her that crying brings mom or dad running, so when that doesn't work she'll ratchet up the intensity until she realizes that things have changed.
After she can handle you being in the room but not giving her attention, move to the doorway, then just outside the doorway, then out of sight but within earshot will start to work. If she fusses after you are already out of the room, wait about 10 minutes before you respond, and respond by poking your head in the doorway, reminding her that it's time to sleep, and walking away.
Why this works
Kids get separation anxiety for one of two reasons (or a combination of both): being away from you (the parent) is new and different and they don't know how to respond, or they are afraid of something (rational or otherwise).
The answer to both is experience. The only way to learn that being away from Mom and Dad is okay is to be away from Mom and Dad. Taking it in small bites makes it easier for your child to digest and they seem to learn more quickly and easily that way then when you try to make them go "cold turkey".