There is no doubt about it. Baby's first birthday party is more for Mom and Dad than for baby. Yes, a one-year old is now old enough to grasp the concept of fun and attention, but realistically he would not care whether his party was comprised of Grandma in a party hat or a roomful of dressed-up toddlers playing games.
I am not trying to suggest you should not have a party to celebrate the first anniversary of baby's arrival. What you should do, however, is lower your expectations.
1. Do not expect baby to care about the presents: Young children are notoriously more enchanted by the boxes presents come in than the gifts themselves. Add practical presents like clothing and books to the mix and you'll be lucky if you can convince the little one to rip the paper off all of the loot. If however, you have invited some three or four year old cousins or family friends, they will be more than happy to live vicariously and do the unwrapping for baby.
2. Do not expect baby to stay neat and clean: The one thing baby is likely to enjoy immensely at his party is the cake. So, even if your precious one has stayed tidy until cake time, it is virtually impossible for him to get past the cake, frosting and ice cream unsullied.
3. Do not expect baby to stay happy/awake/interested: Babies have their own agendas. It is quite probably for baby to fall asleep or have a giant gas bubble-induced tantrum at the height of the festivities.
Once you understand all of this, if you still want to have a party on the big day, by all means go ahead. Plan favors and games for the bigger kids and put out a gourmet spread for the grownups. Take pictures of baby burning his finger on his first birthday candle and enjoy.