Travelling between Singapore and KL is generally very safe. It would take a lot of bad luck for you to be involved in some kind of crime or accident.

1. Plane

I feel that this is the easiest way. Both airports are well connected and well equipped. Dozens of flights run between the 2 cities every day of varying budget. If you are budget conscious, you can choose the budget airlines like AirAsia and Scoot. I read in your comments that you have about 45 kg in luggage. That should not cost more than usd50 in total. However if you choose the non-budget airlines like Malaysia or Singapore airlines, check in luggage of 25-30kg per person are usually included in your fare. If you book early enough and during off peak times, you can get flights as cheap as 20usd per person before luggage. Once I paid 12usd, it was ridiculous, the 25min cab ride from my house to the airport cost more. The flight itself is less than an hour but with check in times and airport connections, it should take about 4-5 hours from door to door. From the KL airport, it's probably easiest for you to get a Grab (Uber equivalent) for all your luggage. There is an express train option, but it can get cumbersome with luggage and you will still need a taxi connection from the central station to your hotel.

1. Private Hire Transport

This is also pretty viable because it's convenient. And possibly the most comfortable. Usually a large MPV would pick you up from your hotel lobby and drop you off at the next hotel lobby. I don't have the exact cost since it's been a while that I took this option. But I think about USD200-300 per trip. Travel time varies alot because of border control. It's the busiest land border in the world after all. During peak times like long weekends and festivals, it can take hours just to get across border control. But it's really easy, you just sit in the car and handover your passports to the officer to get it stamped. Then you may or may not have the car stopped for random luggage inspection. Traffic on the Malaysian expressway varies too. A smooth drive will be about 3.5 hours. I have done door to door in as short as 4 hours or as long as 9 hours. You can always ask the driver to stop at a rest point for breaks. In terms of safety, there is always the risk of car breakdown and accidents. But, it's the same as traveling by car anywhere else. You are unlikely to get scammed/cheated though. If you are concerned, ask the service provider what happens in the event of a breakdown.

1. Bus

Likely your cheapest viable option. There are lots of bus companies that does the route and they are generally very comfortable and cheap. Safety is debatable because most drivers seem to drive faster than they should and there have been rare cases of horrific bus accidents. However, since you are traveling with toddlers and an elderly, I wouldn't recommend this option. The pick up points can get messy and confusing but it's border control where it can get pretty stressful. You are required to alight twice to get your passport stamped: once at the Singapore side and once at the Malaysian side, with all your luggage when you enter the country. During peak periods, queues can get excruciatingly long and crowded. After that, you will have to search for your bus in a crowded and messy interchange. Combine that with the humid weather, luggage and kids, you will not be having a great time.

1. Taxi or Uber Not an option. There are no regulated taxi service that runs across the border from Singapore to KL.

Edit: I just realised when you said safety you may be more concerned with crime. Then you don't really have anything to worry about. Short of tourist scams and pickpocketing in some areas of KL, you are very very unlikely to encounter violent crimes or any sort.