Strong polarization of the ion distribution in liquid electrolytes subjected to potential differences exceeding the thermal voltage, V-T = kT/e, produces a hydrodynamic instability termed electroconvection at ion-selective interfaces. Electroconvection is desirable in some situations (e.g., electrodialysis) because it promotes mixing in a stagnant electrolyte layer, enhancing the ion flux at a fixed potential difference. It is undesirable in others (e.g., electrodeposition of metals) where early experiments show that convective fluid rolls associated with hydrodynamic instability in bounded electrolytes produce preferential metal deposition at localized regions on an electrode. Such localized deposition drives the family of morphological instabilities loosely termed dendritic electrodeposition. We experimentally investigate the effect of ultrahigh molecular weight polymer additives on the onset conditions and physical characteristics of both instabilities. Direct observations of electrodeposit morphology and tracer particle motions used in tandem with indirect electrokinetic measurements reveal that even at moderate concentrations, the polymer additives have a large beneficial effect in extending the range of electric potentials where stable electrodeposition is observed. Additionally, we report that at polymer concentrations above the entanglement threshold, high molecular weight polymers impart elasticity to liquid electrolytes which dampen electroconvective flow at a cation-selective interface but have at most a minimal effect on the bulk ionic conductivity of the liquid.