If "safety" is defined by modern UL/CSA/NEC regulations/codes, then no antique clock or vintage electric appliance is "safe" to run. Put the clock on display and go to Wal Mart and buy a Chinese quartz clock that runs on a 1.5 V battery if you want to maximize safety.
However you can make these old clocks reasonably safe by taking these steps.
1. Inspect the clocks internal wiring and electrical components like the coil and make sure there is no deteriorated wiring. Coil insulation can usually be improved by re-wrapping it with a good tape like 3M #27 Glass Cloth Tape. You can also use triple wall, adhesive lined large diameter heat shrink tubing.
2. Replace any internal wiring that shows signs of degradation. Modern THHN wire will be superior to the old rubber insulated wire.
3. All the clock's internal wiring should be anchored so there is no flex or movement inside the case. (strain relief, tie wraps, etc)
4. The power cord needs to be in good condition with strain relief where it enters the clocks case. You can even source modern cloth covered cord to preserve the vintage look.
5. Those vintage plugs weren't very good and can decay with age. Consider replacing them.
6. To take it one step further, upgrade to a 3 wire electric cord and tie ground (Green) to the metal frame of the clock and use a GFCI protected outlet.
