The formula for this is rather complex. It's based on interest for bank reserves held with the central bank, and it seems that only new deposits will be charged the negative rate.
We're not talking about a straightforward single rate on overnight lending, as is the case for say, the federal funds rate in the US, or the overnight rate in Canada.
Naturally, the dollar has surged compared to the yen, and the dollar continues to look like a safe haven by comparison. But only by comparison, of course, since central banks, partly due to collaboration, and partly due to internal politics, are engaged in a race to the bottom.
Japan has long been leading the race to the bottom, however, since its overnight rate (the Mutan rate) has been under one percent since about 1996:
It's been pretty close to zero most of the time since 1999. (Since I can't find a good source for historical targets, this graph is not of target rates. Its of observed rates, which nevertheless reflect the target rates.)
The observed rate over the past five years has been around 0.06% to 0.09%. (Not to be confused with 0.6% to 0.9%).
The target rate has been at 0.1% since 2009, and thus less than half of the US target rate (0.25) during that time. The sheer length of time that the rate has remained near zero is what's especially notable, however. (Graph from this source.)
Following the gospel that 2% price inflation will solve one's economic problems, the BOJ has been desperate to get price inflation up above zero, where it has usually been for the past decade, except for a surge in late 2014 and early 2015 as Abenomics intensified.
The Japanese economy has continued to weaken, and apparently things have been bad enough to cause the BOJ to follow in the footsteps of the European Central Bank, and do "whatever it takes."