Escalating tensions with Russia sent Ukraine's currency tumbling to a record low against the dollar Tuesday, a move that could threaten the stability of the country's banking system and raise the prospect of losses for bondholders.

The sharp decline this month ends a period of relative tranquility in the country's financial markets. A mid-April interest-rate increase by the Ukrainian central bank and a $17 billion bailout from the International Monetary Fund later in the month had soothed investor jitters. The bailout was...