This is a store brought basil that I planted outside during the spring. It's been growing big and lush all spring/summer and just recently the leaves are getting brown spots and turning yellow. It gets full sun from 1 - 5ish

I water once every morning. Sometimes at night also if it's a hot sunny day.

Is it because of the temperature drop?
Is it getting to much water?

I'm in Zone 7a

Basil don't like cool weather , may be that , everyday water seems like a lot but maybe you are hotter there

I think the plant is getting old sometimes they drop lower leaves when starting to shoot up flower spike .

Those brown marks do suggest temperature. If it has dropped even one night below about 50 degrees, especially with wind, they'll react that way. Looks like that possibly happened maybe 7-10 days ago or so? The new growth looks quite healthy. For the rest of the season (not much time left, unfortunately) see if you can give it some more sun, preferably in the morning. Reduce the watering, too, as the days get cooler.

Can basil survive winter outdoors? Or should I save a few stems and move them indoors til next Spring.

No they wont winter over,They are one of the first plants to get frosted to death
It is usually easier to start new basil from seed in the spring than to try to keep indoors unless you have grow lights or a very bright south window.

. Some people do (try to) grow basil indoor during the winter, usually with poor results without grow lights. Basil is an annual: it germinates, grows, flowers, seeds and dies in one year. Before it gets to 45-50 degrees at night, you can clip several stalks and put them in water and they'll stay fresh for maybe two weeks for culinary use.

I do have a full spectrum grow light indoors on my propagation station. I will try that.

Will this apply to rosemary also?

That's doable. I've only once wintered a potted rosemary indoors and by spring it wasn't very happy, but it did survive in a sunny spot without grow lights. Rosemary is happiest outdoors in full sun and lean, dry-ish soil, but It's a pretty tough plant. The biggest threat to its health is over-watering. I've never used lights. Hopefully someone can jump in here who has done it.

Rosemary freezes up north, Arp is the most cold hardy in zones 6 thru 10. I hear it will survive a 30* freeze but not for extended time. Indoors keep the rosemary temps in 55* to 80*F ranges. It will need a minimum of 11 hrs of light indoors- 12 preferable. Basil, well I grow African Blue to survive my heat, but 32* takes it out. Usually you harvest and dry before frost bites it. The African Blue basil doesn't have seeds so we do nurture cuttings thru the winter that occasionally make it in warmer winters. Not sure about other basils, I don't fight the natural life of the plants and most don't like indoors- if you do try to keep your basil thru the winter, invest in a pot that is 18" deep and drains well- they have long taproots and don't like their feet to stay wet.

My 5'x6' Arp thrives in sustained temps below 30 degrees most years. It blooms in the snow. A potted rosemary needs protection.

Feb 22, 2019

Do you have it protected from wind on any side Zoe? Oh and snow doesn't hurt as many plants as ice or wind.

She is protected on the west and somewhat less so to the north. She was a 3" hardware (possibly grocery) store purchase 8+ years ago when I planted her in unamended sandy soil, not realizing how big she'd get—I identified her as Arp several years later. I've cut her back several times since then, some years quite radically, and she keeps growing and blooming like crazy with virtually no help from me. I give her deep soakings when I think about it, throw on some compost once a year or so, and have always kept her mulched, mostly with dead grass and leaves. The winters have become somewhat milder since I first moved to New Mexico. For the first 5 years or so, we could rely on at least 2 to 4 weeks at least of temps dipping into the teens and many below freezing, because I remember shaded soil being solidly frozen into early spring. And coffee inside my car in an insulted cup freezing during the days. Haven't had that for the last 2 or 3 years.

Anyway, more than you probably wanted to know! The rosemary I overwintered indoors one year was a prostrate variety in a pot. I'd lost some potted hardy lavenders and salvias the year before to freezing, so I didn't want to chance it.

Mine live outside all year, I have prostate and Tuscan Blue...neglected them this year except to pour 2 or 3 gallons water every 2 weeks or less into the 19" hi/19" wide pot they are in. I am hoping to get enough rain this week to accomplish some waiting chores I have around the yard.
Yes, grow lights for indoor plants, I have some spot lamps I put on tender plants- one is blue &red LED lights on a timer, the other is also a clamp lamp but full spectrum and both are on timers. I love them.