Let's compare the total size of circulating BTC to US dollars. According to coinmarketcap.com, the total value of all Bitcoin is 250 billion as measured in US dollars. Throw in Ethereum, Bitcoin Cash, IOTA, Ripple, Monero, Dash, Bitcoin Gold and Litecoin, and you're at 350 billion US dollars. (Bitcoin makes up the sizable majority of all cryptocurrency.)
Now compare to the total amount of US dollars. This one is a little harder to measure because what counts? When you keep a BTC in your wallet, that's just exactly one BTC. But if you keep $10,000 in a bank account, the bank usually loans some percentage of your money out to other people. Economists use varying "money supply" measures. In order of most to least conservative, there's M0 (just circulating paper money and coins), M1 (M0 plus demand deposits), M2 (M1 plus savings deposit and money market funds), M3 (M2 plus large corporate cash balances basically). Then you have near money-like instruments like foreign exchange reserves, treasury bills, short-term corporate paper, foreign currencies closely pegged to the dollar (like the Chinese Remnibi).
Let's just use M2 to get a reasonable lower bound. Currently that value is 13 trillion. So we can say that cryptocurrencies currently represent less than 3% of the total value of circulating US dollar and dollar like instruments.
To devalue the purchasing power of the dollar by 10%, at least 10% of the savings in M2 would have to shift to cryptocurrencies. (And most likely much more than just M2). That would represent an aggregate flow of 1.3 trillion. That would be a gain of 371% from current cryptocurrency capitalization. I.e. Bitcoin at $61,000.
And that's even assuming all the hypothetical new money flow comes from dollar savers only (and no European, Asian or Middle Eastern investors). Not to mention that not all Bitcoin investment is coming from money savers, a lot is coming from demand for speculative assets. In many ways the marginal Bitcoin investor is probably substituting for stocks and commodities. Not CDs and money market funds. More than anything, I'd say its gold and silver prices that probably are impacted the most by Bitcoin mania.
Another way of thinking about Bitcoin is figuring out which fiat currencies it actually sizes up to. Using the above methodologies, taking M2 money supply, all cryptocurrencies put together are still smaller than the Turkish Lira. (Which is about 400 billion USD).
TL;DR Cryptocurrencies at their current size are a pittance compared to major fiat currencies. To meaningfully affect the value of currencies like USD, EUR of JPY, Bitcoin will have to rise in value substantially compared to its current market capitalization. That doesn't mean it won't happen eventually, but current USD savers probably shouldn't be sweating unless the BTC bull market keeps running for a long time.