Hi Mike -
As you can see from the attached screen capture from the SBB timetable, you'll be there very near the end of the Summer/Fall hiking season. Most of the mountain transport does not run after October 22nd.
As Fall proceeds into Winter, many hotels and tourist services take a holiday in late October and November, to do maintenance and give the staff a break before the ski season which starts in early- to mid-December. Also, the risk of enough snow to block the high trails increases, although it is still low in early October. The plants can give a bit of color in the Fall, too. Some is visible in the attached rather old images.
The attached image of the Eiger and Tschuggen was taken during the last week of September in 2009. We had a beautiful sunny day for the walk from Männlichen to Kleine Scheidegg. If you do it, walk toward Kleine Scheidegg. The gradual approach to the Eiger Nordwand ( north face) makes the walk much more impressive. It is also slightly downhill in that direction - from 2230 meters above sea level down to 2061 meters. That is near the top of the list for an easy superbly scenic walk high in the Alps. So easy that is it sometimes nicknamed "The Grannies' Walk."
I have also attached a full resolution crop from the Nordwand image, showing hikers ( just barely visible on the "edge" through the film grain in that rather old 2004 image). The purpose is to give you a sense of scale and the overwhelming size of the Eiger. If you can expand the full size lower resolution image in a photo editor, you'll just barely make out those hikers as a few specks in the full image.
As the weather in the Alps can change quickly and is unpredictable more than 24-48 hours in advance, it is wise to stay in the region for a few days, as you have chosen to do.
When you go through Basel, try to find time to look at the Tinguely water sculpture ( fountain.)
Search with Google for "Tinguely water fountain Basel"