I have a suggestion for getting a feel for how the number system works, which might even convince them that it is a useful thing to know how to do.
Step 1: A method of counting
Take a pile of small objects (beads, lollies, blocks, pieces of spirali pasta, Go counters -- whatever takes your fancy -- though anything that naturally joins together might have a slight advantage). Then say you are going to count how many there are by a new and creative method.
First you'll group them into pairs. There may or may not be one left over but that's ok. Next you'll group your pairs into pairs, making groups of four. There may or may not be a pair left over but that's ok. Now you'll group your groups of four into pairs, making groups of 8.Now group the groups of 8 into pairs, making groups of 16.Now group the groups of 16 into pairs, making groups of 32.If you haven't chosen too many things in your pile it means that now you'll have to stop because there are no pairs of groups. Of course if there are more you'll just keep going on pairing up the biggest groups.
Now how many have we got? Well let's start with the biggest group. We have one group of 32, one group of 8, one group of 2 and a single object. So that's 43 things. As a way of counting it was quite efficient because we weren't actually counting at any point, only pairing things up.
At this stage you can give everyone a pile of things and they can do the pairing up themselves and figure out how many objects they have.
[You can also imagine this being done with the students themselves -- get them to pair up, then pair up the pairs etc. Of course some of them might be a bit miffed at being the ones left out at the end.]
Step 2: A new way to represent numbers
You should be able to say at this point that in fact every number can be grouped like this because they can see that all their piles have been effectively split into pairs progressively. So that means it's a neat way of representing what number you have. But there's an even neater way to do it and here it is.
List the sizes of the piles on the board/document camera/screen: 32, 16, 8, 4, 2, 1
Now underneath say you'll write how many of each pile you have. In our example you have 1 0 1 0 1 1.
Ask them to do this for their own piles and get some or all of them to write their numbers and their list of piles on the board.
At this stage they should be able to see that every number has a representation as a string of 0's and 1's. Ask them if it's possible to get anything other than 0 or 1 (the answer is no, but you want them to be sure).
Then you can say that this is the binary representation of a number. Each digit is how many groups of that size you get when you successively pair things up.
[It may be worth doing this process for other sized groupings, like grouping into 10's, which they may be familiar with from earlier years.]
Now you can give them various binary representations and ask them to tell you what number they represent.
Step 3: Adding binary numbers
Now pair up your students and get them to add their numbers together and find out what the new binary representation is. See if they can do this without having to start from scratch.
Hopefully they hit upon the idea that they can join any groups that are the same size into bigger groups, and continue this process until there are no more. They'll probably do this organically with no structure, and after discussion you can practice by always starting with the smallest piles.
Then you can see how this works on paper with the binary representation.
I think this method could work quite well to get them started on binary numbers, though including some other activities involving counting in binary with the fingers of your hand (as others have described) wouldn't go astray.