Downloading collector is free.
But I guess you want to sync with ArcGIS Online so you'll need at least an ArcGIS Online account/user
A 'free' ArcGIS Online account comes with each ArcMap license so you might already have some.
Additional arcgis Online licenses can be purchased in bundles of 5. See http://www.esri.com/software/arcgis/arcgisonline/purchase
But, strictly speaking, there are no license costs on Collector as such afaik