Airline personnel also need to be trained, or better trained, to be more compassionate in how they handle people, he said, be it demonstrating sympathy when problems arise or simply being specific when asked about delays, saying “20 minutes” instead of just “a few more minutes,” which creates uncertainty and increases frustration.

主句 + be it demonstrating sympathy when problems arise or simply being specific when asked about delays, ...（让步状语从句）

他说，航空公司的员工也需要接受培训，或者接受更好的培训，以便在处理人际关系时更加富有同情心，无论在出现问题时为了表示同情，还是在被问及航班延误时为了明确地说......

这是古英语遗留下来的一种倒装式虚拟语气，be 动词用原形。在正常情况下，本句应该是: whether it is demonstrating...or simply...。

这种倒装式的虚拟语气，常用于if 条件句和 whether，(even) though引导的让步从句。网友的句子，是省略了连词whether的倒装句(后面有论述)。现代英语，只有非常正式的文体才用。不过，在美国英语中，这种虚拟用法经常见到。可以用正常语序，省略连词后用倒装语序。以下句子是正常语序：

If it be right, do it boldly. 假如觉得是对的，就大胆地去做
If this be the case, he deserves you. 当真这样的话，他的确配得上你。
Leave out my name from the gift if it be a burden, but keep my song. 假如我的名字成了负累，就将它从这礼物上拭去吧，但请保留我的诗歌。
If this be the price of the child's sympathy, I cannot pay it. 如果这是换取孩子同情的代价，我是不能支付的。
Whether he be rich or poor, I will marry him all the same.
无论他是富还是穷，我都要嫁给他。
◆如果省略连词，句子要倒装：
Be he rich or poor, I will marry him.
不论他是否有钱，我都要嫁给他。
Be she poor or rich, I will take her as my wife.
不论是贫穷还是富有，我都会娶她为妻。
She exhibits the idealism common to many advocates of sustainability, be it in food or in energy.
她展示了许多可持续发展倡导者所共有的理想主义，无论是在食品还是能源领域。
She enjoys watching dance, be it ballet, national minority dancing or any other sort of dancing.
(= whether it is ballet...)
芭蕾舞也好，民族舞也好，随便什么舞她都喜欢看。
来自《现代汉英综合大词典》
Speak frankly, I pray you, be it for life or death. " 
(= whether it is for life or death)
我请你坦率地讲出来, 不管我是该活还是该死。”
来自英汉文学 - 红字
Be it a painting, a photograph, a poem or a novel, if you created it, you own it and it’s the copyright law itself that assures that ownership.
(= whether it is a painting...)
不管是一张画，一张相片，一首诗或是一本小说，如果你创造了它，你就拥有了它，而它的版权法本身就保证其所有权。
Home is home, be it ever so homely.
(= even though it is ever so homely)
家就是家，即使它是多么平凡。

网友这个让步从句，可以有3种表达方式：
1. 主句 + be it demonstrating sympathy when problems arise or simply being specific when asked about delays, （省略 whether 倒装，虚拟语气）

2. 主句 + whether it be demonstrating sympathy when problems arise or simply being specific when asked about delays, （保留连词 whether...or...，虚拟语气）

3. 主句 + whether it is demonstrating sympathy when problems arise or simply being specific when asked about delays, （保留连词 whether...or...，陈述语气）

【注】demonstrating sympathy 和 being specific，都是动名词作 be 的表语，不是现在分词，不是进行时。

再举一组例子：

不管他是富有还是贫穷，我都要嫁给他。

Whether he is rich or poor, I will marry him.
Whether he be rich or poor, I will marry him.
Be he rich or poor, I will marry him.