Kanishka Sukumar, a consultant who works in midtown Manhattan, says his parents often stash money in gold for safekeeping. But the 24-year-old thinks there is a better place for his savings.

"In times of financial turmoil, I would prefer to see a portfolio full of bitcoin,” said Mr. Sukumar, who holds about one-third of his wealth in the digital currency.

Bitcoin is gaining traction with modern-day gold bugs who question the stability of paper currencies or worry about protecting their savings during the next financial crisis.

A recent survey of 3,500 bitcoin users found that one in five hold the cryptocurrency because they don’t want banks or the government “controlling my money,” according to data provider CoinDesk. That was the second-biggest reason for holding bitcoin, behind owning it as an investment.

Since its inception seven years ago, bitcoin has shared an appeal similar to gold’s for those who want to hold their money outside of central authority. But as its dramatic price swings have eased, bitcoin has started to look like a more viable store of wealth to some of its fans.