There are a few skills you can take to reduce supply demand. One of the very first I prioritize is Field Repairs 3; this is crucial, especially if you want to salvage a lot of ships rather than buying/building them. Combined with Safety Procedures 3 which cuts the negative effects of D-mods in half (and also eliminates supply cost from emergency burn), D-mods change from something you hate to something you love. Combine that with Fleet Logistics 2 and Efficiency Overhaul hullmod, and you basically don't need to worry about maintenance any more. At that point the primary thing costing supplies is CR recovery after battles. This is very important - which ships you field should be based on deployment cost. Also, it's nice to have a salvage ship in your fleet; that can make you mostly self-sufficient in terms of supplies recovered from battles more than compensating for recovery cost.
The other things that can drain supplies are:
1) Salvaging ships. Never, ever salvage (aka "recover") a ship you won't use. If you do so accidentally, either mothball, scrap, or set the ship to "no repairs" immediately. Field Repairs 3 reduces the cost of salvaging ships.
2) Hanging out near stars. Or in neutron star systems. Or especially, in solar flares. When you are in a corona or flare, it will display in your status bar, and your per-day supply consumption will go up by like a factor of 15, from 6 to 100 for example, and bleed you dry really fast - plus, it's hard to maneuver due to the solar wind pressure so you'll stay there longer than expected. Don't even bother investigating derelicts in a solar corona unless you really want to recover that particular ship because you can't buy it. Research stations are worth investigating though.
3) Hyperspace storms. Don't fly into them. Often they will shoot you like a pinball into more hyperspace storms, and each one can cost a lot.
4) Sending too many ships into combat. Only fight with the minimum number of ships you need to have an easy win. Too few, and they get damaged or destroyed, or run out of peak operating time and start costing CR (which means, more supplies). Too many, and you waste supplies on post-combat recovery. I do recommend the "Automatic Orders" mod which will make ships retreat when their peak performance time is over, before their CR starts dropping (which costs supplies). You can always send in more ships later if your initial batch is insufficient.