I think the answer to this questions depends whether you are using a grid and miniatures or similar to handle combat, as opposed to a descriptive combat.
Having looked at the Player's Handbook, there isn't a clear RAW answer to your question. On page 195 it talks about "grabbing or wrestling with" the target - grabbing can be done at arms length, whilst wrestling implies wrapping your arms around it. In both instances, a character is in physical contact with the target.
Area of Effects (line, cone, cube, sphere, cylinder p.204) are not defined by squares as if using a grid, but by length, width and radii. Therefore, the AOE can occupy just part of a grid square as well as an entire square.
If using a grid (p.192), then the grappling rules don't specify you have to enter the target's square to create the grappled condition. Using miniatures and a grid, the two figures just have to be within reach (typically adjacent) of each other. Therefore, if the grappler moves the target into the AOE, wholly or partially, then it is affected whilst the grappler is unaffected, as represented by one miniature being within and one without the AOE.
If not using a grid, then it is not so clear. Grappling and being grappled requires physical contact, therefore I think RAW means the grappler is also affected by the AOE because we are dealing in absolutes (you are either in or out of the AOE, as you either take damage or you don't). There are no rules which specify proportionate damage, e.g. you are only 75% in the AOE therefore you only take 75% of the damage. As the grappler has to have physical contact with the target, and the target has to be in the AOE, it therefore takes the damage as well.
To conclude, if you are using a grid, RAW means the grappler can place the target in the AOE and remain unaffected. If not using a grid, RAW means the grappler is affected as well as the target.