Does an enchantment ability that gives -1/-1 to opponent creatures resolve before other abilities can be used on a 1/1 entering the battlefield
Your opponent cannot sacrifice them before they die. The -1/-1 ability is a static ability. It is always in effect, does not use the stack, and will kill off all */1 creatures before anyone can interact with them. 

A static ability such as "Creatures your opponents control get -1/-1." is always in effect:


  604.1. Static abilities do something all the time rather than being activated or triggered. They are written as statements, and they’re simply true.


That means creatures this applies to enter the battlefield with the -1/-1 effect. The game never sees a 1/1 creature, it enters as a 0/0. As soon as the spell or effect that put that creature on the battlefield has resolved, the game checks for state-based actions (A set of rules that is checked almost all the time), because the active player would gain priority (the right to take actions).


  116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
  
  116.5. Each time a player would get priority, the game first performs all applicable state-based actions as a single event (see rule 704, “State-Based Actions”),[..]
  
  704.5. The state-based actions are as follows:
  
  704.5f If a creature has toughness 0 or less, it’s put into its owner’s graveyard. Regeneration can’t replace this event.