EPS8266 ESP-01 Blue led stays lit unless 3.3v logic divider is bypassed, works when reconnected
A wiring diagram would help. 

Also, measure the 3.3v regulator's output, especially under load; 9v batteries are best in low-drain long-life applications like smoke detectors. Another supply, perhaps Arduino's 5v supply, might be a better source for the 3.3v regulator.