Calculate Base Resistor in a NPN transistor switch
You find this by first determining the collector current requirements of the transistor, then working backwards to find the base current, then the resistor to support the base current.

To find what current the transistor needs to switch, remove the transistor and close the switch with a ammeter instead.  That will tell you how much current the string of R2 and the buzzer require when the bottom end of the buzzer is grounded.  That's the collector current the transistor must be able to support.

Actually, that's the average current.  Depending on the type of buzzer, this current may be in bursts that are higher.  However, the capacitor will help to smooth this out.  I'd want the transistor to support at least twice what you measure given no other information.

It's not totally clear, but let's assume this is the 13 mA figure you gave.  We will therefore want the transistor to support 26 mA minimum.

The base current is the collector current divided by the transistor gain.  You get the gain from the datasheet.  I'll use 50 as the minimum guaranteed value in this example.  It's your job to look up the real number.  (26 mA)/50 = 520 &micro;A, which is the minimum required base current.

Figure the B-E junction drops 700 mV.  Since you're starting with 3 V, that leaves 2.3 V across the base resistor.  Lose the diode in series with the base.  I can't even guess what you think it's purpose is.  Now it's just Ohm's law, (2.3 V)/(520 &micro;A) = 4.4 k&Omega;.

Since you found you need a much lower base resistor, something is rather different from this example, even considering the gratuitous diode you had in series with the base.  I suspect the peak current required by the buzzer is much more than you think.

Even with the extra diode, you should be getting 7.3 mA base current.  The maximum possible collector current is with the transistor saturated, say 200 mV, and the remainder across the resistor.  (8.8 V)/(100 &Omega;) = 88 mA.  The maximum required transistor gain is (88 mA)/(7.3 mA) = 12.  That is well below what a 2N3904 can do at that current, so something in your setup is wrong.  You may have misread a resistor value someplace, or as Jim Dearden mentioned, have C-E of the transistor flipped.  They still work like a transistor that way, but usually with lower gain.