Not sure about the visible scale for labels in ArcGIS Online, but here's an Esri Blog Post on adding a halo to the label text.
Suppose you want to create halos for your web map labels. First, start with your annotation feature class in ArcMap. You then use the Feature Outline Masks tool (in System Toolboxes > Cartography Tools > Masking Tools), selecting your annotation feature class as your input layer. When you used this tool to create the original labels you specified a Margin of zero and set the Mask Kind to EXACT. Since you now want to create a halo, you set the Margin to greater than zero (in map units) so that feature outlines will be created as buffered polygons around the annotation. For the labels in figure 1 a margin of 50km was used and the Mask Kind set to EXACT (figure 2). The resulting polygon feature class can then be exported to a shapefile and added to your web map in ArcGIS.com in the same way as the original labels. The effect you see in figure 2 was achieved by placing the halo shapefile underneath the labels shapefile in the web map layers and applying 50% transparency.