Can you pay life as part of a cost if you only have 1 life remaining?

Yes, you may pay life as a cost even if you only have 1 life remaining. Paying a life cost (such as {B}{B} with Phyrexian mana, or activating a ability like Dark Confidant's trigger) is not prevented simply because you have 1 life. The payment is made, the cost is paid, and the spell or ability goes onto the stack.

However, state-based actions are checked immediately after the cost is paid (and continuously throughout the game). CR 704.3 specifies that state-based actions are checked after each player receives priority. If paying the life reduces you to 0 or less life, CR 704.5a states that a player with 0 or fewer life points loses the game as a state-based action.

This means the spell or ability you just activated or cast will not resolve — you lose the game before priority is passed and before the stack resolves. The cost was paid, but you do not get the benefit.

Example: You are at 1 life and cast Gitaxian Probe by paying 2 life (Phyrexian mana). You pay the 2 life, dropping to -1. State-based actions are checked; you are at 0 or less life and lose the game immediately. Gitaxian Probe never resolves.

One important exception: if something prevents you from losing the game at 0 life (e.g., Platinum Angel is on the battlefield), you can pay life freely below 0 without immediately losing, and your spell or ability will resolve normally per CR 704.5a's dependency on that effect.