Early soundcards used lower resolution audio, as well as more simple sound generators/synths, such as basic sine, square, or sawtooth waves.
In GarageBand, you can add the "Bitcrusher" effect on your instrument track to get a more lo-fi sound. The older computer speakers also didn't have as much bandwidth, so you can add some EQ filtering as well.
This combo worked pretty well for me, try this:
1. For your instrument, go to the Edit tab and use the Analog Basic module as your sound generator. Try one of the subsettings that has the word "square" in it's name, those seem to be more basic synth sounds closer to a simple square wave. "Bright Square Bass" sounded pretty good to me.
2. Add the "Bitcrusher" effect. Open up the manual settings for it and set the resolution to "8-bit" and the sample rate reduction to "4x"
3. Open the "Visual EQ" effect settings. Expand the detailed settings. Set the Bass EQ to around 200 Hz at -24 dB, the High Mid to a 10 dB boost at around 1500 Hz, and the Treble to around 3000 Hz at -24 dB.
Another option, since GarageBand doesn't have an actual synthesizer, is you could download one as a free AU instrument and use that within GarageBand. One I've come across is
Automat. Once you download and run the install, it will show up as a sound generator module in GarageBand. In the Automat settings window, just use the default settings and set the oscillators to Saw, Sine, or Square, or a combination of the three. That should get you the nice simple synth sound you're looking for.
