Since these are 1 year olds, the party is really for the adults, right? hmmmm...
You've got a tight budget. Are you feeding people? or is this just for your little bundle of joy?
Of course get some bubbles and blow them in the air-everyone loves this. Cheap.
Favors:
Get a stack of white wash cloths and a bunch of small rubber duckies and place one small rubber ducky wrapped up in the wash cloth and tie a big yellow ribbon around it to give as party favors. You can get the washcloths and rubber ducks at walmart or $1 store.
Food:
Not sure if it's a finger food snacks or a meal bday
You can always do chips and dip. Veggie plate and cheese and meat tray. Easy and cheap. If it's a meal...it's gonna cost. How many people? It's always expensive to feed people. Can you have beer there and call it suds in a bucket? Just a thought.
You have to have soda and punch there too.
Decor: I'm stumped on decor.
Are you crafty and artistic? Make a cake and draw with icing a tub and a rubber duckie on it for sure.