Hmmm, let's see... First: Have you checked the tape connection for the recorder? Tape/Record Out on the Technics should go to Analog In on the Sony - and Tape/Playback/Monitor In on the Technics should be connected the Analog Out on the Sony. Checked?
Ok, next thing to check for: The output on the Gemini you ought to use for recording purposes is the Record Out - that's the line output of your mixer supposed to provide the mix for recording purposes. The other ones (Master & Booth) have additional volume controls and are meant to connect power amps (or powered monitors) for the pa speakers in the dance hall as well as some monitor speakers in the dj booth or in another second listening zone (e.g. the bar area...). And it's correct to connect that Record Out to the Aux In on your Technics, 'cause the phono stages in the mixer already provide a line-level signal - that means the phono signal is already amped by factor ~100 and RIAA (re-)equalized (and that's why it won't work with the Phono In on your Technics - using that, you'd perform the amplification and RIAA (re-)equalization twice...).
Now, next step is to produce a good signal for recording. On your mixer, you'd best use channel 2 and 3 for connecting your turntables - 'cause then you can use the level indicator of your mixer to control the output level (use the switch on the meter to switch from master output level to channel 2 & 3 level control!). Now put on a record and search for a loud part - with the slider in the marked standard position turn up the gain control so that you get some 0 - 3 dB reading on the level indicator). Now we should have a good signal on the Record Out. On your Sony, you should now switch the input for recording to Analog In - then you should be ready to record, using the level control for adjusting the record level.
That's about it - i.e. that's the way it should work.
Greetings from Hannover!
Manfred / lini