The ongoing campaign is a product, governed by the same rules that constrain any kind of product development - time, scope, and resources. And these constraints translate into sandbox terms through the freedom, size, and level of detail you're putting into the sandbox. We know this subconsciously; the dungeon is a popular adventure locale precisely because it constrains freedom and size, making it easy to develop.
My wife and kids are even familiar with the term sandbox; they both have characters in the Skyrim game, which they tell me is an "epic sized sandbox", and are frequently encouraging me to give it a try and pick up some ideas from it. (I'm usually too busy for video games, but it sure looks impressive). The wise Google tells me that Skyrim was developed by a team of 100 developers over a couple of years of development. How does a home DM compete with a project with those kinds of resources behind it?
You can't break the law of physics. If it takes 10 minutes for you to write an encounter, and you want to make 6 of them, it's going to take an hour of game prep. But there are some simple techniques we can use to make sure you're focusing your energy on the right 6 encounters for the next game session. Sandbox gaming doesn't have to overwhelm the DM; here are three simple techniques to get the game started right and keep it going once launched.
Kick it Off with an Adventure
Choice requires options, which in return requires information. Nothing drags on a game session like a gigantic information dump, and you don't want to start your new campaign putting the players to sleep. Introduce the campaign with a simple adventure and let them learn about the larger campaign world later, through play. They'll thank you later.
I have no problem starting off a new campaign with the player characters outside the gates of the ruined manor outside of town, possessing their starting gear and some rumors about a lost treasure. There will be plenty of time for them to learn about all the other places and things to do later on.
Why it works: "Show don't tell" and starting with some early action are tried and true narrative techniques from other media that work fine in the RPG context, too. Restricting the first game to a single plot hook or adventure site puts a cap on how much prep you need to do up front and launches the game right into an adventure.
What's Next Week
A good sandbox practice is to reserve time at the end of each game session to ask the players what they'll do next week. Based on the current night's activities, the players already have an idea what they want to do next… continue exploring the dungeon, go to the city and find a Remove Curse, follow the treasure map into the mountains.
Why it works: The sandbox structure gives the players a ton of freedom to make their own plans; this can be daunting to prepare in a vacuum. By giving you some advance notice on next week's plans, you can focus your weekly efforts, and keep preparation to a reasonable window. (IT geek side note: For all you project managers and Scrum folks out there - this is basically weekly sprint planning for your campaign. Your players are the product owners reprioritizing the backlog as input to your sprint.)
Leverage the Tools
Plans change, and while the players indicated last week they were going to explore the dungeon, running into that petrification monster early on in tonight's session changed their priorities; now they're off to the city, mid-session, to seek a cure. EEK!
Smarter folks than me have written primers on building fantasy sandboxes with just enough of a skeleton structure to support improvisation if things go in an unexpected direction. I like the simple "tagging" approach used in the Sine Nomine books, Red Tide or Stars Without Number, coupled with a hex map and lay of the land. ACKS has a whole chapter on campaign creation as well. The approaches are not mutually exclusive.
However, that sandbox skeleton needs to be supplemented with some tables - wandering encounters, for instance, to help create game content on the fly.
Why it works: Tables are a direct, experiential way to define your setting - you're building the details of your setting while creating your own tools at the same time. Tables can cover wide geographic areas and maximize setting scope with the least effort. They're the ultimate tool.