Remittances without high fees.
I think as entropy becomes very inefficient to produce (difficulty multipliers increasing on crypto as the network expands...) it will become more clear that the expensive part about sending money internationally is trusting all parties involved.
You pay VISA/MASTERCARD a 3% fee because you trust them to protect your account in the case of fraud. Moneygram takes 2% of all transcations over $900 because you trust them to move money from your account to the recipients account without losing it.
Even in computer security, you have to put your trust in something. Somebody picked cryptography and high numbers of CPU cycles.
Anyway, you asked what problems crypto could solve, and I guess my answer is, maybe it will cost less to send money around the world. As it turns out, there's not necessarily a way to make it free.