I feel your pain; I have a son who turned one the beginning of this month and my daughter will be three in June. I'm fortunate that my daughter is very understanding and realizes that sometimes her brother needs me to do things for him. From day one, I've tried to include them in each other's activites. Yes, I realize a newborn isn't going to do much, but what I mean is that when he was awake I always kept him near so that she would understand he was always going to be a part of things. And I've always let her help me with things that needed to be done, right done to feeding him when he started solids.
Even with the best planning and best intentions, things can still be a mess. I don't think I've ever let one of them cry while I'm interacting with the other (diaper changes, toilet time, getting dressed; during those types of things perhaps). I do tell my son that mommy is trying to read a story or his sister is trying to go to sleep (when I'm trying to get them both to take a nap), when he does have a fit trying to get my attention. This is more for my daughter's benefit to make her realize that I still think her feeling and what we do together are still important.
When I'm doing other things, getting dressed, drying my hair, cooking meals, etc. and he throws a fit b/c he wants to be held, I do let him cry and talk to him explaining what is happening and why I can't pick him up. Yes, it's difficult, but I need him to start understanding that he can't throw a fit everytime he wants something (which my daughter never did).
When my son is napping and my daughter is awake, I try to give her as much one on one time as I can. I try to give her extra time at the end of the day also b/c my son goes to sleep first. My daughter sometimes reverts to baby like behavior, which can be annoying, but I remind her that her brother acts that way b/c he has no other means to communicate, but she has words that she can use. This sometimes works.
Lengthy, I know and not full of too many suggestions, but I wanted to let you know that you are not alone. For my son this started within the last month or so and they say separation anxiety peaks from 12 -18 months, so my fingers are crossed, although it may be a long 6 months.