The degree of precision will dictate the degree of ease with which this can be done! You can't assume that Google Earth imagery is georeferenced correctly, otherwise you would just grab a screenshot, take it to Inkscape, scale it, and trace the items of interest onto a new layer. You then have a map, of sorts. You could also take your image, go into the field with it, identify a number (number determines final accuracy) of marker (ground control) points, survey them as accurately as you feel meets your precision requirement (with your phone, or with RTK DGPS), import the whole lot into a GIS package, georeference and correct the image, then trace your mapping onto a new layer. Then you have a better map. Whether that's as accurate as you want for the level of effort you require is a subjective decision only you can make! Good luck.
Following the answer from Neil, I would suggest you test the feasibility of google-earth by comparing it with georeferenced data within a GIS software. You can do this on ArcGis or QGis by using specific plugins which allow you to import google-earth (plus Microsoft-Bing in QGis) images into the GIS desktop.
Nonetheless, bear in mind that image quality and xy precision tends vary from place to place. Choosing study sites which are consistently covered by the same set of images tends to help. Best of luck!
You can use the "polygon" tool in the google earth and digitize the river network that you need to map, and just save it as .kmz file. Then you can open the file in the ArcGIS as ADD data-open the ". kmz" file that you saved from the google earth, choose the placemark polygon file of the river network that you digitized. Best of luck....
By the way, in case you don't already know, QGis is free, and very powerful. So in terms of costs, your outlay on this exercise is mostly in time. A friend of mine is the founder of Planet Labs - over the next few years (5-10?) - we should have access to high resolution, high quality, iteratively collected visible spectrum data for nominal costs. Like Google Earth on steroids - bring it on!
Maps cannot be directly or precisely drawn from Google Earth. To map any area precisely you must first download the SRTM Data of version 4.1 (free downloading) from the internet. Then use the software of either GlobalMapper (recent versions, 14.1 or 15) or GIS (you have to purchase) to establish a précised and colored topographic map for your area , then download the World imagery directly from Global Mapper (included in all versions) or Google Map (included in version 15) in order to obtain a satellite (geomorphologic) map for the same area. Also you can obtain contour lines, water shades, cross sections, and ground elevation of original surface as well as the height of artificial surface ( dimensions and height of Buildings, etc , could be obtained from version 15 of GlobalMapper). If you have a software for the Geology,mineralogy or Urban distribution or any other data (either on line or private files) of your area you can also download them on your map. This is not a publicity, but I like to inform you what I already used to draw my maps.