We will be traveling through Switzerland in early June. When we get to Interlaken, we plan on sating in Murren for 3 nights. Can we park our car safely in Interlaken for that amount of time?
Most likely, yes, but why not just leave the vehicle in the large car park in Stechelberg, by the gondola that you ride up to Mürren?
Bypass Interlaken - just keep on driving to Lauterbrunnen. There's a large parking garage in Lauterbrunnen where you can park your car. It's across the street (linked by underground tunnel) to the gondola that will take you up to Grutschalp where you'll change to the tiny train all the way to Murren.
Alternately you can also continue driving through Lauterbrunnen to Stechelberg - about 5 more minutes. There's a gondola station there (as well as parking). The gondola takes you through Gimmelwald and on up to Murren (opposite end of Murren from the train station) in the first paragraph above.
So there are 2 ways up to Murren. Either is good.
But really the best way to travel to Murren (and all over Switzerland) is to use the wonderful public transport system.
I agree with the others. Forget Interlaken and park in either Lauterbrunnen or Stechelberg (Lauterbrunnen would be my choice). From there take the cable car up to Grütschalp, and then connect with the small mountain railway (which will be waiting).
You could also park in Stechelberg and take the Schilthornbahn to Gimmelwald and then Mürren. Either route will work but you'll have to know which one you'll be taking when you buy tickets. The Lauterbrunnen - Grütschalp - Mürren trip will be specified as "BLM".