Is there any danger of forming explosive reaction products during the electrodeposition in Diethylene glycol?
Diethylne glycol reacts readily with perchloricacid, however the oxidation and hazard potential of perchloric acid depends on its concentration in side the solvent.

If concentrations are high enough and more than 10 percent perchloric acid formation is expected with 100 percent reaction yield, then it can be explosive.

What actually happens that as soon as diethylene gylcol meets with perchloric acid, diethylene glycol decomposes and some dangerous peroxides formed. These are very reactive to water, so if you can provide very dry and inert atmosphere it should be safe but you should give extra care to make sure that you do not have humidity and water both in your chemicals and in the atmosphere.

If you are going to do the experiment and use milimolar range chemicals then just do it, because you will not be able to form enough perchloricacid to get problems.