There are several different ways to approach this depending on the exact brand and model of washer that you have. If you look on the quick browse list in the community – created wiki in the smart app section and check the “alerts and notifications“ list you should find several of them.
Basically there has to be an event that smartthings can recognize to use as a trigger for the notification. In most cases these will only work if you are using the smartthings classic mobile app, but if you happen to have a Samsung smart washer, then you would have to be using the “smartthings (Samsung connect)“ mobile app.
1. The washer tells you
If the washer itself is a smart device and either has an IFTTT service or is a Samsung washer that works with the connect app, you can just use that as the trigger for the notification.
If the washer has its own app and you have an android phone, you can also probably use a combination of Tasker and SharpTools To capture the notification event to the app.
If the washer can send an email or SMS, which some can, then you can capture that through IFTTT.
But all of these are based on the washer itself knowing that it is done and being able to contact you.
2. The washer vibrates while it is running and stops when it is done.
This is a typical method if you have a washer which is at least five years old. These washers generally have a drum which spins through their entire operation. This can usually be detected by any vibration Sensor that works with smartthings, including the smartthings brand multisensor. However, many newer washers have multi cycle operation where the drum spins for a while and then stops, and then starts up again later. This can cause the sensor to go off too early unless you use Webcore to create a much more complicated detection algorithm.
3. The washer draws more power while it’s operating than when it’s done.
This method only works with some washers. But if the washer does draw detectably more power while operating, you can add an energy monitoring device. I personally don’t like this as well because in order for it to work you generally have to have nearly continuous polling of the energy usage on that outlet, and while I know people like this Idea conceptually, it’s just not a really good match to a mesh network like zwave or zigbee home automation. You can end up flooding the network with Messages for something you aren’t even using. I believe at one point @tgauchat said that over 3/4 of the events from actiontiles users were just for energy monitoring. 
. But it is a popular method.
4. If the washer has a dial that rotates while it is operating, you can use a battery-operated Contact sensor to recognize when the dial reaches the stop position
A couple of community members have done this. I like it a lot. It doesn’t send unnecessary messages, the device itself is pretty inexpensive, you shouldn’t get any early alert misfires, there’s just a question of the aesthetics. And of course it only works on specific models.
5. Acoustic alert
There aren’t very many sound sensors that work with smartthings, but there are a couple if you go through IFTTT, Including some cameras that have built-in sound sensors. So if the laundry room is in an otherwise quiet area, you might be able to get an alert when the “laundry done” signal goes off. I’m less optimistic about this working reliably just because so many other sounds can set it off, but it will be an option for some people.