As a tourist, you technically only have a few transportation choices, taxi, BTS (skytrain), MRT (underground) and tuk-tuk (not much experience so I won't touch on this here).
Thais love kids more than you can ever imagine, so do trust me that bringing kids will likely be your trump-card. In BTS and MRTs, we've never had issues with seats, no matter how crowded the carriages are. Even when I bring Noah on the trains myself (and he's already 3), there will still be many locals (young and very old) 'fighting' to offer us their seats.
Do take note also that if you have a baby/toddler in tow, whether or not you have a stroller/lots of bags to carry, you will be helped by the security personnel stationed at every turnstile at a train station. They will definitely help you through with ease via a separate gate at the side so you don't have to rush your way through the barrier with all your 'load'.
Taxis will be a more touchy subject. There are real issues with taxi-drivers trying to earn more from tourists, choosing destinations and refusing to switch on their metres. However, with a baby in tow, even the most opportunistic taxi-driver might decide to give you a friendly ride, in order to give you some assistance.
Assuming we do not expect any child-friendly facilities at road-side stalls and establishments without air-conditioning just like most other countries, let me zero in on the typical restaurant in a shopping mall.
Many established restaurant chains will at the very least, have child bowls and cutlery, and a number will have dedicated kids menus, but you will have to be prepared when it comes to baby chairs. In most restaurants, the most they can provide might only be chairs for slightly bigger kids, who do not need safety latches or protective tables in front of them. If you have a baby/toddler less than 2 years old, you will need to hold them while you/they eat.
Staff at most restaurants are very friendly to kids though. If they are free, they might even help you carry your baby or play with your toddler, something not common in many other societies.
If you are looking at the normal roads, the answer is NO. The pedestrian pavements are not designed to be suitable for strollers or prams. If you insist to bring yours along, my suggestion is to stay in 2 areas, the zone between Chit-lom and Siam BTS stations, and the zone connected to Asoke BTS station. These zones provide a conducive skywalk above the road that connects you to the malls and BTS/MRT stations in the vicinity where you can use your strollers with ease. However, do take note that thus far, only a select few BTS stations provide lifts, so be mentally prepared to navigate many flights of steps with your stroller.
An easier way is to carry your baby/toddler in your carrier if you are used to it. This way, you do not have to worry about the stroller outside the malls. In fact, many malls now provide a stroller-rental service. These include Terminal 21, Mega Bang Na and Paradise Park. I have seen strollers in Emporium's customer-service office as well, my take is that if you are at a proper mall, you might want to ask the information counter whether a stroller is available for rent.
- Availability of nursing rooms
In most malls, family/handicapped-friendly toilets are available, but nursing rooms might not be. To be safe, if you need a nursing room, choose to go to a mall with a departmental store (which will surely have nursing rooms in their kid/baby sections). Popular departmental store chains in Bangkok will be names like Central, Zen, Isetan, Robinson, Tokyu, Emporium and Paragon. This means that popular tourist malls like Platinum Mall, Siam Centre, Siam Square 1 and Terminal 21 (though it has a Robinson near it) do NOT have nursing rooms.
- What about essentials like diapers, milk and baby food?
This should not be a big concern. Most big brands are established here. Furthermore, they can all be found in most 7-11s, which are everywhere, so don't worry about this. In fact, there are many local high-quality ready-made baby food/snacks which are available in the supermarkets. Look out for brands like Peachy, Wel-B, Picnic Baby etc.
- Weather and Air Quality
Depending on where you come from and where you stay, you might have different interpretations to this concern. Put things simply, Bangkok is a little more seasonal than countries like Singapore and Malaysia. The best time to come is surely at the end of the year, when temperatures can drop to a low of 18 degC in the morning. This usually happens from end-December to January. It doesn't rain during the cool season as well. The rainy season usually starts from June to August, so this is not a very conducive period to visit. The hottest time is during Songkran, from April to May.
Compared to 'Garden Cities' like Singapore, there are hardly any trees in the Bangkok. Coupled with the massive jams in the tourist areas most of the time, the air quality is expectedly poorer in comparison. However, if you are used to visiting other cities like Shanghai, Beijing, Jakarta etc., this should not be a concern as well.
- Healthcare for the little ones
In town, good hospitals are not difficult to find. If anyone in your family needs medical treatment, remember the names Bangkok Hospital, Bumrungrad Hospital and Samitivej Hospital. These are top-class private hospitals with translators available for almost all languages I can think of. They are of course, not cheap, but compared with a good private hospital in your country, you might find them providing an equal/higher value of service.
So, in summary, is Bangkok baby/toddler friendly?
If you ask me, my answer is a great big YES! If you take note of the pointers above, be mentally-prepared for some situations different from your home country and be less adventurous in your sight-seeing visits (and I suggest, book a hotel connected to or very near a BTS/MRT station), you might understand that even though Bangkok's 'hardware' is still lacking, the 'heartware' is already in place.
With the increased number of sincere smiles channeled in your direction because of your precious little ones, you will realise by the end of your trip that bringing your baby/toddler along is actually a blessing in disguise.