John,
What you have drawn is the
emitters going to the resistors. So an LED connected between the other end of the resistor (cathode) and +5 (anode) would get current. No need for a base resistor.
However, this is not the best way to do it. The transistor in this configuration is acting as an emitter follower. There will be 0.6V (plus or minus some mV) between base and emitter (resistor). Then the LED will drop between 1.7V (ordinary red LED) and 3.5V (newer LED types) or maybe as much as 3.6V for blue LEDs. That leaves as little as 0.8V for the resistor, less if your 5V supply is on the low side.
So say you calculate the resistor for 20mA at 5V - 3.6V - 0.6V = 0.8V. That's 40 ohms, so you use 39 ohms. Say it turns out that the supply is actually 4.4V (the allowable minimum for a USB charger). Now you have only 0.2V for the resistor, and your LED current will be 5mA, not the intended 20mA.
Because you have so little voltage across the resistor any variation in the factors that control it will give proportionally larger variations in the current.
If you reconfigure as shown in the diagram (this is my first post, so I'm not sure if I'm doing the diagram properly), the transistor will "hard switch" and saturate at around 50-100mV. So you have gained 0.5V or so extra for the resistor. Doesn't sound like much, but it's a 63% increase, reducing the sensitivity to variations in voltage by that much.
View attachment 33703
In general, driving a LED that needs 3.6V off a 5V (nominal) supply is never going to be a precision operation, once all the possible variations are taken into account.
That answers your question as asked.
Now, is there a better way? Most likely. I am not intimately familiar with PIC, and you didn't specify the chip type. However, most micros these days (I say these days because I've been doing this for nearly 50 years) can drive LEDs directly. And modern LEDs are so bright that they will burn your eyeballs on 5mA. Just go through a resistor directly to the LED. Get the PIC data sheet and find the output voltage drop at say 5mA. Get the LED data and find the voltage drop at 5A. Subtract both from 5V, divide by 5 and you have the resistance in KΩ.