MUMBAI—The rupee is the worst performing currency in Asia this year, flirting with a record low on worries that India is headed for problems as oil prices remain high.

The rupee has declined 5.3% against the dollar this year and has underperformed against almost all others. Only those of Argentina, Brazil, Russia and Turkey have done worse. Most currencies have fallen against the dollar since the start of the year over a strong U.S. economy and rising U.S. Treasury yields.

The...