Arduino RGB ledstrip control 2 separate power supplies
Yes, there is a problem. There is no return path for the base drive. You are hoping to pull current from the base of each transistor but there is no return path to the 12 V PSU so it won't work. You need to connect the negative of both PSUs together and to the microcontroller's ground. This would be much easier to see if you had drawn a proper schematic rather than a hybrid wiring diagram.

The second problem is a "high-side fail".



Figure 1. The protection diodes on most logic chips creates a sneak-path to positive supply. This will keep the PNP transistor permanently turned on and may damage the chip. Image source: GPIO high-side driver fail.

If If  VSS > VMICRO then the output protection diodes built into the chip provide a sneak path for the base current. The e-b junction of Q1 will be forward biased and current will flow through it, R1 and D1 to the micro supply. Q1 will turn on and LED will light and will not switch off. If the current is high enough the microcontroller will be damaged.

The linked article (mine) shows a fix for that if a high-side driver is required.