for perimeter calculation you just right click on basin shapfile and then select attribute table and then create a field with a specific name. Then right click on this column and select calculate geometry tool. when you select this tool a window will appear that say which geometry you want to calculate. you should choose Perimeter and press Ok button. After this you can find the calculated perimeter..
For basin length calculation you need longest flow path and basin area. By dividing basin area per longest flow path, basin length can be obtained. For calculating longest flow path a suggest you GeoHMS or ArcHydro extensions. If you need more help, don't hesitate to contact me.
You can add column to attribute table with title perimeter, then right click this column and press calculate geometry and units of perimeter.you will get the desired perimeter
You will add column to attribute table with title Perimeter, then right click this column and click on calculate geometry and units of perimeter.you will get the perimeter.