The rich rule over the poor, and the borrower is a slave to the lender. ~ Proverbs 22:7
I believe the answer to this question is both “no,” and “yes,” depending on the situation.
Let’s start with “no.” There are two main reasons that I don’t think we can deem the act of borrowing to always be sin:
A) THE BIBLE DOESN'T SAY THAT BORROWING IS A SIN.
That is, unless you count Romans 13:8. However, you’d have to read Romans 13:8 out of context to come up with that interpretation. The verse just before it makes it clear that sometimes Christians do “owe” things to people, and therefore they ought to pay them. And so, verse 8 seems to be a riffing off of that verse, saying something to the effect of, “Make all the payments that are required of you until you have no more obligations, except for the giving of love, to which you are forever obligated.”
B) THE OLD TESTAMENT REGULATES LENDING AND BORROWING.
Exodus 22:25 commands God’s people to not charge interest when they lend to the poor, and Deuteronomy 24:10-13 gives regulations for what a person was and wasn’t allowed to do when accepting collateral for a loan. It seems that God would not instruct His people on how to lend if it was always a sin to borrow.
However, just because something isn’t always a sin doesn’t mean that it’s never a sin. There are some things that are wrong in some situations that aren’t wrong in every situation. (For instance, it’s generally true that it’s not a sin to date a girl named Katie. However, it is wrong to date a girl named Katie if your wife’s name is Brenda.)
Here’s another verse that I think could be applied to the idea of debt:
Do not be conformed to this age, but be transformed by the renewing of your mind, so that you may discern what is the good, pleasing, and perfect will of God. ~ Romans 12:2