Only you know what is right for you. You may wish to ask someone to observe in your absence. Be aware that, for safety reasons, your veterinarian will probably not allow you to be touching or holding the animal when it is put down. Discuss the procedure in advance so you know what to expect. Make arrangements for the prompt removal and disposal of the body.
Many municipalities have ordinances prohibiting or restricting burial.
Removal to a rendering facility or pet crematory may be required. Explain to members of your family, especially children, in sensitive but honest terms, why the decision was made to euthanize the horse. Allow yourself to grieve. Finding a support person or group to talk to can help you work through this difficult period. If the horse is insured, notify the insurance company in advance of the euthanasia so that there are no problems with claims.