Sail rather than fly, travel in late February or March, take a lower cabin grade, and stay flexible about which ship and which date. Those four together move the number more than anything else. What doesn't work any more is waiting for a deep last minute deal, which used to be the budget traveller's strategy and has largely stopped being one.
Flexibility is the part that pays. If you can say "any ship, any 10 to 12 day Peninsula departure in March, cheapest cabin", we can work with that and find you something good. If you need a specific ship on a specific week with a specific cabin, you're paying for that precision.
Lower cabin grades are the least painful saving on this trip. You'll be ashore, in a Zodiac, on deck or at a lecture for most of your waking hours. A porthole cabin is not a compromised Antarctic experience.
Solo travellers should ask specifically about shared cabin options and dedicated single space, which changes the economics considerably compared with paying a single supplement.