Lots of places work, but anywhere in the area bordered by Constitution Ave to the south, Nst NW to the north, the Potomac to the west, and 4th st NE to the east is great.
Also, if you want to save on parking, you can also park overnight outside the city and take the metro in when going/coming back, more details on that here: https://www.wmata.com/service/parking/ (go to multi-day parking tab) . Wiehle metro's garage would be most convenient from Wisconsin, is incredibly safe and only $5 total if staying less than 10 days.
The question actually wasn't about how to get cheap parking, it was how to get "security" parking, which I don't think is that easy. I really depends what that means, if it means some garage with a guard there 24/7, that isn't very common, I don't think. Any suburbs on the north or west side of DC will be safe enough. DC is safe enough if you stick to the NW quadrant. There aren't a whole lof ot hotels in the SW, anyway, there are some near the Navy Yard but just look for NW in the address to make it easy. Even then, I'd avoid any hotel real close to N Capitol St, if you find one. There aren't a lot, but a few around H St to NY Ave, I thinkl.
BUt I think the Hyatt in Bethesda may use the public parking garage anyway, which is pretty cheap but nothing particularly special in its security (nothing at all). I don't know, it could be they have their own underground garage, you'd have to ask themj.
Maybe you can say what that means to you. It should be cheaper in the suburbs, no matter what. I think a lot of DC hotels just have valets who take it to some nearby public garage, but not totally sure. I think the Omni up in Woodley Park has its own garage and that's a nice area. It isn't cheap, though (the garage I think is $55 per day, but the hotel rates aren't that bad). I just stayed in a Fairfield Inn and Suites elsewhere, and it had its own underground garage but it was completely open all night long, so nothing especially secure about it, I suppose. Of course that means clients had in/out privileges whenever they wanted, which was great. The Hampton Inn near the White House has valet parking, you'd have to ask them what that means, I don't know, but that's a good location. The Hilton Dupont Circle (really several blocks north but that makes driving easier) had its own huge garage, I've parked there since it's a big convention hotel. They say it is "secure" but I don't really know what that means, you'd have to ask them.
You might feel more comfortable in the VA suburbs than in DC proper, and they really aren't that far out. We usually stay at the Residence Inn Merrifield - in Falls Church, VA - about a 20 minute drive to downtown DC outside of rush hour. They have a secured underground garage, but we feel fine parking inthe outdoor lot as well. It's adjacent to a strip mall with several good restaurants.
The most secured parking in DC for the general public would be at the Ronald Reagan Building. It is a federal facility, so the armed guards will check your trunk before allowing you in. You can stay in nearby hotels and park there.
Any parking garage will be as secure as you can get. Which there are really very few open air parking lots left in the city so then it just comes down to price and convenience.
Hotels in the CBD use a mix of private garages and nearby places they contract with.
I think the Reagan building is a good idea for security, but it is big, so you could do stuff in there without being that obvious. But you do have security to get in, as I recall. I wouldn't say any parking garage is as secure as any other. Metro has a lot of theft from their garages, for example. More theft FROM cars then theft of car. The issue is metro doesn't use a lot of surveillance cameras in their lots or garages, and the attendants are only on duty during working hours. Some private garages are closed up at night, those would be more secure, I'd say.
Yeah, that's true of most metro garages but the one at Wiehle goes down to 7 stories underground so theft is much harder, they also have an option of a private garage there (Blvd parking, allows overnight for $5 a day) which is also very secure.