When working with logic gates and transistors, which is more important to pass through them for them to work, a high current or a high voltage?
A high level at a logic gate is a high voltage. Gates generally draw little or no current at their inputs.
If the logic gate in your diagram uses one of the most common technologies, that is CMOS or TTL, then it cannot be connected directly to that bipolar transistor without risk of damage. The transistor base will not rise more than about 0.7 V above the emitter, and will draw too much current from the logic gate output. There needs to be a series resistor between the logic output and the transistor base to provide a current drive to it. Typically you will aim to supply 3% to 10% of the collector current to the base terminal to make sure the transistor is on.
If the bipolar is replaced with a FET, then that will accept a high voltage directly, and will draw no current from the logic output.
There are many more options that could confound answers to general questions like these. Some logic gates have open collector outputs which cannot source current, and so need a pullup resistor, but then can be connected directly to a bipolar. FETs have different threshold voltages, some newer ones work well with 5 V logic levels, most FETs especially older ones really require much more than 5 V on the gate to work properly. There are other logic families than TTL and CMOS, and their logic levels would need to be designed with properly.