USB serial adapters come in a number of flavours.
1) 5V. This would pass through the 5V to the ESP8266 which is way over the esp8266 and likely to damage it.
2) Basic 3.3V. These provide 3.3V from the USB serial chip which have limited current capability and are unlikely to satisfactorily power the esp8266
3) 3.3V with separate regulator for external devices. These will probably work fine.
Better is to avoid the ESP-01, which in my view is a nasty device, and use a much more friendly Wemos or NodeMCU which include their own usb serial and a proper 3.3V regulator