THE FIRST FROST need not put an end to the enjoyment of fresh vegetables from the garden. When frost is predicted, some can be protected for a night or two and enjoyed during the Indian Summer weather that often follows.

Use old sheets, table cloths or shower curtains to cover the plants in early evening to fend off frost during the night. Tuck the covering around the foliage and fruits to keep out the cold night air.

When the temperature rises above 50 degrees F. in the morning, remove the covering. Put it back in the evening if frost is still in the weather forecast.

Tomatoes are not the only vegetable to consider covering, according to Ted Torrey, Burpee vegetable breeder.

Perhaps you have zucchini squash just starting to bear nice fruit or cantaloupes almost ripe. It certainly would be worthwhile to cover them and other frost-susceptible plants. You may be able to harvest from them for two or three more weeks.

When a forecast is for prolonged freezing weather the growing season will come to an end. At that time, the best thing to do is to pick as many immature vegetables as you can and store them for ripening (see Vegetables, LIVING section, Sept. 27, 1981).

Whitish-green tomatoes can be pulled from the vines to take inside to ripen at 55 to 60 degrees F., or the vines can be pulled up and hung in the garage for the fruit to ripen.

Dig white potatoes intended for storage when the tops are mature. Do this early in the day to avoid field heat. Don't expose them to the sun for any length of time. Put them in well-ventilated containers such as half-bushel slatted baskets.

Store the potatoes where the temperature will be between 60 and 80 degrees F. and leave them to cure for four days. For permanent storage, move them to a cooler place where they can be kept at 45 to 50 degrees F. They should not be kept where they will be exposed to artificial light for extended periods, or where last year's crop was stored unless it is thoroughly cleaned. Properly stored, they should keep well for about four to six months.

Spinach and parsley usually revive after a heavy frost. Carrots, salsify (oysterplant) and parsnips can be left in the ground and dug anytime the ground isn't frozen.

Brussels sprouts and kale really take cold weather. Brussels sprouts almost covered with snow when harvested and cooked had an unusually mild scrumptious flavor. But cook them soon after picking because after being partially frozen, they tend to get soft and spoil if kept above freezing.

Kale often tastes better after frostbite and can be picked from under ice and snow during the winter. Beets and turnips can be stored in a cool dark place, or leave them in the ground and cover with a thick layer of straw or evergreen boughs. Whenever the weather permits, push aside the covering and pull as many roots as you want.

Winter-type onions will keep for several months in a cool dry place. Be sure the onion necks are dry and shrunken before harvesting. This indicates proper maturity and good storage life.

Dig sweet potatoes carefully to avoid bruising, cure them at 80 to 85 degrees F. for about 10 days and than store at 50 to 55. They will sprout in about 30 days at higher temperatures.

Winter type squash and pumpkins can be stored successfully in a moderately dry basement at about 60 degrees F. Put them on a shelf or counter. They could spoil if kept on a damp concrete floor.

Q. We want to start an asparagus bed. Can you give some tips on how to go about it?

A. Asparagus is a perennial, which means it will live for several years without having to be replanted. The best way to start is with 1-year old roots, which can be planted after danger of hard freezes is over in the spring but before growth starts. Select a well drained location for the bed. Have the soil tested at your state university; asparagus does best in slightly acid soil. If your soil is poor, add 5 or 6 bushes of well-rotted manure or compost per 100 square feet.

Q. For the last two years our peaches from dwarf trees were wonderful. This year they have all been moldy. Can you tell me what happened?

A. It was due to a fungus disease called brown rot. Next year, spray with an all-purpose fruit-tree spray, follow directions on the label for mix and application.

Q. I want to dig up my vegetable garden this fall to have it ready for spring planting. Is there any serious danger of erosion of the soil?

Each rain drop that falls may break some particles from the soil mass, which can be carried away by surface runoff. If the ground is covered with snow or frozen for much of the winter, there is not likely to be much erosion.

Q. I saw a beautiful tree yesterday bearing clusters of seed pods resembling nuts, the name tag on it was Paulownia imperialis. Is it easily grown? Can young trees be purchased from nurseries?

A. The Paulownia, sometimes called the Empress Tree, grows rapidly and is generally considered to be a weed tree. It can be seen growing wild in many places. Nurseries seldom seldom stock it, probably because there is little if any demand for it. The tree is easy to grow from seed which mature in mid-to-late fall and should be planted immediately.