well just throw in a resistor. the higher the value, the less current the arduino will draw for monitoring. I dont know how much the arduino would draw without the resistor. I believe its not much. Get out a multimeter and check if you want to know the exact numbers :-)
One thing I neglected in my original response was that I assumed you are using a 5V Arduino. If you are using a 3.3V board then don't connect a 3.7V battery directly.
Assuming a 5V Arduino: Connect Battery - to the GND on the board. Connect Battery + to one end of a 10K resistor and the other end of the resistor to the analog input pin. It doesn't have to be 10K, it can be just about anything from a few hundred ohms to a few hundred K ohms. Use the following formula in your code:Voltage = 5 * Counts / 1023
Voltage(at the pin) = VBatt * R2 / (R1 + R2) andADC Counts = VPin * 1023 / 3.3 <== Replace 3.3 with 5.0 if using 5V system From those two equations we get the following formula which can be used by your software:VBatt = (Counts * 3.3 / 1023) * (R1 + R2) / R2 <== Replace 3.3 with 5.0 if using a 5V system
Based on what I've read you probably shouldn't worry about a capacitor. If you have a lot of noise in your system, then adding a capacitor across R2 (from Analog pin to GND) will filter out some of the noise. The formulas for the filter are a little more complicated, so I suggest a 1uF capacitor. If it is still noisy, try 10uF or ask again, telling us your resistor values and how much noise you are getting.